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ARITHMETIC 


METHODS  AND  REVIEWS 


BY 

W.  H.  BAKER 

AND 

ADELIA   R.   HORNBROOK 


19  14 


Press  of  Eaton  &  Co, 
San  Jose,  Cal. 


CALIFORNIA  EDUCATION 


PRICE,  50  CENTS  PER  NUMBER 


NUMBERS  NOT  EXHAUSTED 

VOL.  I. 

CONTENTS    FOR    NO.    2,    MARCH,     1906 

Editorial,     Margaret  B.  Schallenberger,  Ph.  D 85 

History  for  Seventh    Grade,    Agnes  B.  Howe,  A.  B 87 

School    Gardening,    D.   R.    Wood,  B.   S 126 

Country  Schools  and   Country   Life, B.  B.  Dresslar,  Ph.  D 135 

CONTENTS    FOR    NO.    4,    SEPTEMBER,    1906 

Editorial,     Margaret  E.   Schallenberger,  Ph.   D 199 

Manual   Training    in    Public  Schools, Bdwin  R.  Snyder,  A.  B 201 

VOL.  n. 

CONTENTS   FOR    NO.    1,   JUNE,    1908 

Editorial,     Margaret   E.    Schallenberger,   Ph.   D '.       3 

History   for    Eighth    Grade, Agnes    E.    Blowe,   A.    B 4 

CONTENTS  FOR  NO.  3,  OCTOBER,  1910 

Editorial,     Margaret   E.   Schallenberger,  Ph.   D 141 

Nature   Study   and    Agriculture D.  R.  Wood,  B.  S 143 

CONTENTS  FOR  NO.  4,  SEPTEMBER,  1914. 

(Double   Number,   Price  75   cents) 

Letters  of  a  Supervisor  of  Arithmetic, Adelia    R.    Hornbrook 1 

Arithmetic — Methods   and    Reviews, W.  H.  Baker   31 


(COPYRIGHT,    1914,   BY   W.   H.   BAKER) 


3/7 


<"^ 


PREFACE 


This  manual  has  been  written  especially  to  meet  the  needs  of  normal 
students.  Incidentally  it  may  be  found  useful  to  other  students  and  teachers. 
Besides  methods,  it  presents  reviews  of  subject  matter.  While  the  teacher  who 
has  not  had  methods  is  greatly  handicapped,  the  theory  of  teaching  is  of  little 
value  to  one  who  has  not  mastered  the  thing  to  be  taught. 

The  letters  of  Mrs.  Hornbrook  are  specifically  for  the  student  teachers  of 
the  Training  School.  The  first  sets  forth  the  general  scheme  of  her  work  and 
gives  a  number  of  illustrative  exercises  and  games.  The  second  gives  a  gen- 
eral outline  of  the  course,  which  will  be  fully  amplified  in  the  supervisor's 
meetings  from  week  to  week. 

The  chapter  on  drills  gives  suggestive  exercises  intended  to  help  the  busy 
teacher  and  to  keep  the  pupil  occupied  in  what  will  interest  him  and  cause  him 
to  know. 

The  chapter  on  course  gives  a  skeleton  outline  for  work  beginning  with  the 
fifth  year.     It  is  tentative,  suggesting  an  order  of  sequence. 

A  large  percentage  of  normal  students  have  had  no  arithmetic  since  leav- 
ing the  grammar  grades,  hence  their  knowledge  of  the  subject  is  quite  limited 
and  their  work  more  or  less  mechanical.  They  need  a  firmer  grasp  of  the 
subject  matter  and  the  feeling  of  security  which  comes  with  conscious  mastery. 
The  chapter  on  reviews  is  designed  to  supply  these  needs.  The  arrangement 
is  strictly  topical.  While  there  are  many  simple  exercises  and  problems  the 
work  is  in  general  too  difficult  for  grammar  grade  pupils.  The  teacher,  how- 
ever, should  know  more  than  she  is  expected  to  teach.  The  many  model  solu- 
tions are  intended  to  furnish  good  forms,  which  is  important,  and  incidentally 
to  illustrate  the  best  ways  of  doing  the  work. 

Modestly  yet  hopefully  this  manual  is  offered  to  the  aspiring,  earnest 
teacher.  The  more  ambitious  task  of  presenting  the  psychology  and  pedagogy 
of  the  subject  is  left  for  others. 

W.  H.   B. 

State  Normal,  San  Jose, 
September,  1914. 


786797 


INDEX 
To  Open  Letter  and  First  Five  Years 


Abacus     22 

Accuracy  required    5,  9,   16,  24,  28 

Apparatus   9,  21,  22 

Automatic   knowing    13,  25 

Charts,  representations  of  the  number 
series  up  to  100,  2;  as  used  by  Miss 
Smith,  6;  as  a  basis  for  children's 
discoveries     14 

Chart   of   fives,   3 ;    construction    of,   8,   21 

Chart    of    elevens    15 

Child,   the   shy    7,   8 

Co-nbinations  and  Separations,  first  set, 
18,  27;  second  set,  19,  27,  29;  third 
set,   19,  27,  29. 

Correction    books     28 

Correlation  of  the  tables 23,  28 

Courtesy  in  the  schoolroom 5,  6 

Diagram    of    classroom 20 

Different  ways  of  treating  quick  pupils 
and  slow  pupils. ..  .10,   11,  14,  16,  23,  26 

Discipline 4,    5,  23 

Even    Numbers    18,  25 

Failure,    a    negation    5 

Fifth  Year,  work  of 30 

First    Year,    work    of 21 

Fourth    Year,    work    of 30 

Fractional  parts  of  objects  and  num- 
bers       19,  26 

Games,  plays,  occupations,  3-9,  12,  13, 
22-30. 

Geometric    forms    19,   25,   27 

Individual    Advance     28,    29 

Individual    Tests     23,    7] 

Inventing    Plays    12 

Learning   the    Sight    Series 6 

Learning    the    Sound    Series 4 

Learning  to  Apply  Number  12 


Million    Stick    !.  26 

Movement,    rhythmic,    free 24 

Number    forms     2 

Number    Stories    17,    28,  30 

Parallel   Lines  of  Work 19,  27 

Parents     as     Visitors 4 

Perception     Work     27 

Plans    19,  20,  21 

Play    Spirit    22 

Playing   Leader    7 

Presenting  the  Digits    7,  8,  25 

Prof.    Wm.    James'    theory    of    "brain- 
paths"     16 

Progressive    written    work     28 

Reports    20 

Running  to  a   number 24,  25 

Second    Year,    work    of 26 

Sectioning    the    Grades 21 

Series   Idea   in    Number    1 

Storing  the  subconsciousness,  5,  8,   11,  13, 
14,   25. 

Story  of  a  successful  young  teacher,  22,  23 

Stunts   5,  21,  24,  27 

Table  of  tens  and   its   reverse 23 

Teacher's    tools    21,    22 

Teaching    ahead    '..28,    29 

Teaching    tables    of    multiplication    and 

division 27,    28,    29.    30 

Third  Year,  work  of 28 

Two    periods    of   number   learning,    first 

period,   13,   14,   15,   16;   second  13,   16,   17 

Visualization     2,    11,    13,    15 

Voluntary   effort    3,   5,   7,    13,  21 

Written    multiplication     28 

Written    subtraction     28 


I. 

INTRODUCTION 


AIM. 

The  manner  of  teaching  arithmetic,  and  its  content  are  determined  in  a  large 
measure  by  the  purpose  for  which  it  is  taught  and  the  teacher's  conception  of  its 
educational  value.  If  it  is  taught  merely  as  a  tool  subject,  only  such  topics  will 
be  considered  as  are  of  direct  service  and  the  method  will  be  such  as  will  bring 
about  the  master}'  of  these  topics  ,with  the  least  possible  expenditure  of  time  and 
energy.  On  the  other  hand,  if  only  the  culture  vahie  of  the  subject  is  taken  into 
consideration,  the  method  of  study  and  the  way  in  which  it  is  presented  will  be 
deemed  of  more  importance  than  the  topics  themselves. 

Arithmetic  is  a  tool  subject  and  this  fact  makes  it  imperative  that  certain 
topics  be  included  in  the  coursej^ancl  taught  with  that  degree  of  thoroughness  tliat 
will  enable  the  pupil  to  do  the  work  readily  and  accurately^  The  form  of  work 
should  be  such  as  to  enable  the  learner  to  get  the  desired  result  as  quickly  and 
correctly  as  is  consistent  with  clearness. 

Arithmetic  as  a  tool  is  used  by  the  pupil  in  pursuing  his  studies  in  the  ele- 
mentary schools.  For  this  purpose  he  needs  to  understand  the  fundamental 
processes  as  applied  to  whole  numbers,  common  fractions,  and  decimals^  In  the 
secondary  school  arithmetic  is  needed  in  the  mathematical  work  in  chemistry  and 
in  physics.  For  these  the  pupil  needs  a  thorough  grounding  in  common  frac- 
tions, decimals,  and  percentage.  Business  life  demands  a  knowledge  of  percent- 
age, interest,  mensuration,  and  lumber  measure  in  varying  degrees  according  to 
one's  occupation.  Since  arithmetic  is  seldom  studied  outside  of  the  elementary 
school  these  topics  should  be  included  in  the  course. 

Business  men  have  devised  certain  self  explanatory  forms  and  direct  ways 
of  reaching  results  which  should  be  taught  when  the  proper  time  comes,  instead 
of  the  cumbersome  and  mechanical  forms  usually  given. 

School  time  is  growing  time  mentally  as  well  as  physically.  Whatever  the 
child  studies  should  minister  to  this  growth,  each  subject  or  activity  in  its  own 
peculiar  way.  Arithmetic  as  a  culture  subject  ought  to  be  instrumental  in  the  for- 
mation of  certain  good  habits.  It  ought  to  train  to  a  neat,  methodical  arrange- 
ment of  material,  a  careful  and  thoughtful  study  of  the  meaning  of  expressions, 
and  a  disposition  to  inquire  for  reasons  and  search  for  truth.  Arithmetic  ought 
also  to  train  in  thinking  and  in  clear  and  concise  expression  of  relations,  not  by 
the  repetition  of  set  formulas,  but  by  the  logical  sequence. 

Arithmetic  is  valuable  as  a  medium  through  which  the  pupil  gets  informa- 
tion. Customs  and  forms  prevailing  in  the  business  world  may  and  should  be 
learned.  These  will  be  recognized  as  of  vital  importance  and  will  be  lastingly 
impressed  if  presented  in  living  practical  problems.  Thru  problems  the  child 
can  get  a  saner  knowledge  of  the  commercial  and  geographic  importance  of  na- 
tions and  of  the  problems  of  civilization. 

MENTAL  ATTITUDE. 
Authority — Language  looks  to  usage  past  and  present  for  its  rules.     The 


vi  .  ARITHMETIC 

historian  looks  to  the  records  and  traditions  of  the  past  and  sifts  truth  from  them. 
"I  have  looked  it  up  and  I  find  it  so",  is  the  final  statement  of  Hnguist  or  his- 
torian. 

Respect  for  authority  is  an  important  attitude  of  mind.  It  urges  man  to 
stop  and  consider  carefully  before  discarding  present  laws  and  customs  and  sub- 
stituting untried  theories  in  their  stead. 

Experiment — The  scientist  is  a  questioner,  a  doubter.  He  is  not  satisfied 
with  present  conclusions.  Authority  is  to  him  a  finger  pointing  in  a  direction  in 
which  truth  may  be  sought,  and  he  seeks  to  verify  or  disprove  by  his  own  inves- 
tigation. "I  have  made  careful  examination  and  I  find  it  so",  is  his  tentative 
statement 

To  this  modern  scientific  spirit  much  of  the  world's  progress  is  due. 

Proof — The  element  of  necessity  underlies  the  progress  of  the  mathematician. 
Given  certain  axioms  and  postulates,  his  conclusions  follow  inevitably.  He  may 
examine  his  work  for  error  of  statement  or  oversight,  but  finding  none  he  can  not 
question  the  result.  When  a  truth  has  been  settled  mathematically  there  is  an 
end  of  the  controversy.  "I  have  demonstrated  therefore  I  know",  is  final  with 
the  mathematician.    He  can  turn  his  attention  to  other  problems. 

This  feeling  of  certainty  can  and  should  be  in  the  mind  of  the  student  of 
mathematics  from  the  beginning. 

INTEREST-MOTIVE. 

I  The  educational  value  of  a  study  depends  largely  on  the  amount  of  mental 
energy  called  forth  by  it,  and  this  is  dependent  on  the  interest  with  which  the 
study  is  pursued  and  the  motive  which  awakens  this  interest.  Teacher  and  pupil 
must  co-operate  if  there  is  to  be  substantial  progress  in  education.  A  task  may  be 
assigned  and  the  pupil  required  to  perfrom  it#incited  by  fear  of  punishment  or 
hope  of  reward.  Thru  such  means  the  pupil  may  acquire  knowledge  and  skill, 
but  he  will  have  little  pleasure  in  his  task,  and  the  resultant  mental  growth  will 
be  a  minimum^J 

Use. — Use  is  commonly  considered  the  prime  motive  power^  If  this  use  is  in 
the  future  its  influence  on  the  average  elementary  pupil  is  slight,  f^he  child  lives 
in  the  present  and  for  the  present,  and  the  incentive  of  the  future  usefulness  of  a 
study  supplies  only  an  artificial  stimuluSj^  Present  use  arises  from  present  neces- 
sity, and  can  last  only  while  the  necessity  exists,  hence  its  influence  is  at  best  but 
temporary.  It  emphasizes  the  necessity  for  certain  knowledge  and  serves  as  a 
reason  for  the  study  of  a  particular  topic,  but  it  does  not  and  can  not  furnish  the 
motive  for  the  persistent  study  which  is  necessary  if  there  is  to  be  any  perma- 
nent beneficial  educational  result. 

r^  Inherent  Interest — The  motive  which  will  call  forth  persistent  interested  ef- 
fort is  that  which  comes  from  the  subject  itself  as  influenced  by  the  teacher  and 
the  manner  of  presentatiorhj  To  know,  to  understand,  to  be  able  to  do,  to  be 
skilled  in  doing  bring  their  own  reward  and  awaken  an  abiding  and  increasing 
interest.  Ken,  can,  and  king  are  words  closely  related  in  language  and  in  human 
nature.  He  who  Mens  can,  and  he  who  can  is  made  king;  and  rejoicing  in  his 
power  he  seeks  to  increase  it. 

Arithmetic  properly  studied  affords  its  own  incentive  and  reward.  To  know 
numbers  and  to  be  able  to  handle  them  with  accuracy  and  intelligence  give  genu- 
ine pleasure  to  the  student ;  there  is  a  definiteness  about  the  results  and  a  cer- 
tainty about  the  conclusions  which  give  satisfaction ;  and  one  feels  that  it  is  worth 
while  to  pursue  a  subject  in  which  there  is  a  consciousness  of  increasing  power.  J 


INTRODUCTION  vii 

Freed  from  drudgery  and  grind,  it  is  possible  for  arithmetic  to  be  made  from  the 
first  a  pleasant  and  interesting,  therefore  profitable  study. 

Play — The  recreative  value  of  play  has  long  been  recognized.  Cities  are  re- 
alizing that  play  may  be  made  to  have  a  socializing  and  civilizing  force,  and  are 
spending  millions  in  purchasing  and  equipping  grounds  and  providing  them  with 
proper  supervision.  Play  is  no  less  play  because  it  is  wisely  directed.  We  have 
been  strangely  slow  in  recognizing  the  educational  value  of  play,  overlooking  the 
fact  that  a  large  part  of  our  knowledge  and  skill  have  been  attained  thru  that 
channel.  The  boy,  and  sometimes  the  girl,  learns  to  ride,  to  hunt,  swim,  row, 
and  skate  thru  play  The  farm  boy  learns  to  handle  tools,  drive,  plow,  and  do  a 
hundred  other  things,  thru  play,  and  while  learning  enjoys  himself  to  the  full. 
The  eagerness  with  which  children  pursue  manual  training  and  domestic  sci- 
ence is  due  largely  to  the  gratification  of  the  play  instinct.  Children  do  not 
play  at  a  thing  because  it  is  easy.  From  the  child  trundling  his  wagon  to  the 
football  enthusiast,  real  play  is  real  work  gladly  and  rigorously  performed. 
The  play  spirit  enables  one  to  do  his  best  under  the  best  possible  conditions. 
The  child's  play  is  his  work.  Happy  is  the  adult  whose  work  becomes  his 
play. 

The  pl?y  instinct  should  be  largely  utilized  in  teaching  arithmetic.  The 
child's  desire  to  count  and  his  joy  in  counting,  his  pleasure  in  constructing  and 
destroying  that  he  may  construct  again,  his  inquisitiveness  and  eagerness  in  mak- 
ing his  own  discoveries  should  be  made  to  contribute  to  his  advancement,  to  lay 
the  foundation  for  future  discoveries,  and  to  awaken  an  abiding  interest. 
Thruout  the  course  the  arithmetic  recitation  should  be  looked  forward  to 
with  pleasure  and  keen  anticipation,  and  should  leave  a  memory  of  happy  attain- 
ment and  a  determination  for  continued  effort. 

PHASES. 

In  considering  the  subject,  one  must  recognize  two  distinct  phases  of  the 
work.  These  are:  first,  the  ability  to  handle  numbers,  i.  e.,  to  perform  the  opera- 
tions, addition,  subtraction,  multiplication,  and  division  of  whole  numbers,  com- 
mon fractions,  and  decimals,  and  extracting  roots,  and,  second,  the  application 
of  these  operations  in  concrete  situations.  Tho  usually  taught  together,  these 
phases  are  logically  distinct,  and  methods  which  will  develop  efficiency  in  one 
will  in  no  wise  advance  the  other.  The  abilitv  to  determ.ine  what  operation  should 
be  applied  in  a  given  problem  does  not  come  with  the  ability  to  perform  the  op- 
eration, neither  is  it  learned  thru  a  set  form  of  words.  It  is  a  matter  of  experi- 
ence and  judgment.  Within  the  horizon  of  the  child's  experience,  and  especially 
v.'ithin  the  circle  of  his  interests,  he  seldom  makes  a  mistake.  It  is  when  we  place 
before  him  situations  foreign  to  his  experience  or  in  which  he  has  no  personal  in- 
terest that  he  flounders  about.  Because  of  these  facts  no  textbook  in  arithmetic 
should  be  placed  in  the  hands  of  the  pupils  during  the  first  three  years.  During 
the  first  year  he  should  wrestle  entirely  with  problems  which  arise  out  of  his  work 
or  play.  It  is  not  necessary  that  these  problemis  involve  number.  The  laying  out 
of  a  flower  bed,  the  building  of  a  cardboard  house,  or  the  putting  together  of 
spools  and  box  to  make  a  wagon  will  result  in  growth  and  power.  The  weights 
and  measures  in  common  use  should  be  handled  that  they  may  furnish  material 
for  future  work.  Problems  for  the  second  and  third  years  should  arise  from  the 
work  or  play  or  come  from  the  living  teacher  and  should  have  direct  reference  to 
the  child's  immediate  environment.  In  later  problem  work,  attention  should 
first  be  directed  to  the  meaning  of  the  problem,  what  is  given  and  what  is  re- 
quired. Care  should  be  taken,  especially  when  taking  up  a  new  type  of  problem, 
that  small  numbers  shall  be  used  and  that  the  operations  involved  be  not  com- 
plex.    Forms  of  analysis  are  of  questionable  value. 


viii  ARITHMETIC 

Rationalization-UShall  the  operations  be  taught  as  a  matter  of  pure  memory 
and  drill,  or  shall  there  be  an  appeal  to  the  understanding?  Whatever  method 
is  adopted  there  must  be  much  resort  to  drill  and  much  memory  work  must  be  re- 
quired. The  pupil  must  finally  come  to  perform  the  work;  more  or  less  automat- 
ically; accuracy  and  a  greater  or  less  degree  of  speed  must  become  habitual.  If, 
however,  memory  work  alone  is  resorted  to  the  result  will  be  that  the  pupil  will 
become  mechanical  in  the  extremej^  Mechanical  in  performing  his  operations,  the 
child  will  almost  certainly  become  a  guesser  as  to  what  operation  should  be  per- 
formed. Too  many  will  be  so  under  the  best  cqnditions^^On  the  other  hand 
many  pedagogical  sins  have  been  committed  and^miich  harm  has  been  done  thru 
attempts  at  rationalization.  Authors  and  teachers  alike  have  erred  in  this  re- 
spect; bad  forms  have  been  impressed  and  bad  habits  have  been  encouraged. 

(These  formsN^eing  placed  before  the  child  when  he  is  learning  a  process  im- 
press themselves  upon  him  and  they  become  to  him  a  necessary  part  of  the  opera- 
tion. Even  the  use  OfJ^plints^eans,  counters,  and  the  like)  in  rationalization, 
may  become  stumbling  blocks/^Some  pupils  become  slaves  to  objects,  and  can 
work  intelligently  only  so  far  as  they  can  see  or  image  the  object,  and  when  such 
objective  work  is  impossible  because  of  the  largeness  of  the  numbers  fail  utterly 
or  become  the  worst  of  mechanical  workers.  Others  resort  to  marks  on  boards 
or  papers,  count  the  fingers,  or  make  motions  in  the  air.  A  few  survive  and  be- 
come fairly  good  mathematicians  despite  bad  methods.  In  the  following  pages 
the  attempt  is  made  to  bring  about  an  intelligent  appreciation  of  the  processes 
and  yet  avoid  the  e\ils  referred  to,  for  only  by  so  doing  can  the  work  be  made 
enjoyable.  What  is  learned  as  a  distasteful  task  has  little  value  educationallys^ 
If  must  not  be  expected  that  any  pupil  can  understand  all  the  processes  as  their 
use  becomes  necessary.  The  rationale  of  some  operations  is  too  difficult  for  chil- 
dren at  the  age  at  which  such  operations  should  be  learned.  In  other  cases  it  is  of 
little  consequence.  The  inversion  of  the  divisor  in  division  of  fractions  is  an  in- 
stance of  the  first  kind,  why  to  begin  at  the  left  in  division  is  of  the  second.  Many 
pupils  will  get  little  good  out  of  any  attempt  at  rationalization.  This  fact,  how- 
ever, should  not  keep  the  teacher  from  making  any  such  attempt.  Some  seeds 
may  rot  in  the  soil,  others  may  be  long  in  geniiination,  but  for  the  sake  of  those 
which  grow  we  must  not  withhold  the  planting.  Mr.  McMurray's  rule  is  not  a 
bad  one,  tho  not  without  exceptions :  "It  is  folly  to  sacrifice  the  present  for  the 
future.  Any  subject  that  cannot  be  fairly  comprehended  at  the  time  it  is  present- 
ed should  be  excluded"  (Ed.  Rev.,  Vol.  27,  p.  482).  While,  then,  we  must  not 
attempt  complete  rationalization,  whenever  the  child  of  ordinary  ability  can  be 
led  to  comprehend  the  reason  for  an  operation,  he  should  be  given  the  opportunity. 

'  SPECIAL  FEATURES. 

In  a  few  respects  the  course  herein  outlined  differs  from  others, 
Exposure-Play — In  the  work  outlined  for  the  primary  grades  Mrs.  Horn- 
brook  has  shown  the  happy  way  in  which  play  may  be  made  to  coimt  toward  real 
acquisition  in  arithmetic.  She  has  outlined  a  number  of  interesting  exercises, 
and  these  will  suggest  many  others  to  the  resourceful  teacher.  These  exercises 
build  up  and  keep  continually  before  the  child  a  useful  picture  of  numbers  to  one 
hundred,  the  full  value  of  which  can  be  appreciated  only  by  the  teacher  who  has 
patiently  and  thoughtfully  used  it.  They  bring  the  child  into  the  presence  of  laws 
thinly  veiled,  and  the  pupils  soon  begin  to  find  out  for  themselves 
facts  about  numbers.  Thru  these  exposures  and  thru  play  the  subcon- 
scious mind  is  charged.  By  what  alchemy  it  works  we  do  not  know,  but  we 
know  that  thru  it  the  child  finds  pleasure  and  profit  in  handling  numbers. 
— •  Law — An  attempt  is  made  to  lead  the  child  to  investigate  for  himself  and' 
thru  such  investigation  to  discover  truth.  The  pleasure  of  personal  discovery  is 
great  and  the  habit  of  looking  for  reasons  and  laws  is  a  good  educational  asset. 
There  are  many  useful  laws  governing  numbers  so  simple  that  the  child  cait 


INTRODUCTION 


IX 


grasp  them.  These  are  utilized  and  emphasized.  Children  take  delight  in  certain 
exercises,  such  as  counting  by  twos,  tens  or  fives.  These  pleasurable  exercises 
are  made  to  do  service  in  learning  the  combinations  of  these  numbers  in  addi- 
tion, multiplication,  etc. 

Groups — The  pupils  are  encouraged  to  recognize  certain  group  results  such 
as  the  groups  making  eight,  nine,  ten,  and  so  on.  These  groups  are  emiphasized 
at  the  proper  time  and  the  habit  of  adding  them  as  a  single  number  is  formed. 
Groups  making  nine  should  be  learned  after  the  child  learns  to  add  nines;  in  gen- 
eral a  group  making  a  given  number  should  be  learned  when  the  child  has  learned 
to  add  that  number. 

Ebdng" — The  pupils  are  not  required  or  expected  to  use  a  combination  in  a 
miscellaneous  way  before  it  has  been  well  fixed  in  the  memory.  To  do  so  is  likely 
to  result  in  bad  habits  and  inaccuracies.  To  fix  a  combination  requires  much 
repetition  and  it  ought  to  be  interested  repetition.  This  is  secured  thru  counting, 
decade  work,  and  simple  column  work,  making  use  of  the  eye,  the  ear,  and  the 
sense  of  rhythm. 

Small  Nr.mbers — In  the  work  of  the  first  three  years  no  attempt  is  made  to 
deal  with  large  numbers.  The  pupil  is  taught  to  read  and  write  numbers  to 
10,000,  but  he  has  no  occasion  to  use  large  numbers  at  this  age  and  the  mechan- 
ical handling  of  them  is  postponed.  After  the  child  has  grown  into  a  knowledge 
of  the  principle  of  place  value  he  is  less  likely  to  become  a  machine.  Many  chil- 
dren take  a  delight  in  reading  and  writing  large  numbers  and  in  handling  them. 
Such  are  not  discouraged.  Their  questions  are  patiently  answered  and  their 
feats  receive  due  recognition. 

Eeasons — Rationalization  is  not  sought  thru  categorical  statement  of  a  gen- 
eral principle.  However  apt  such  a  statement  may  be  it  is  likely  to  be  another's 
rather  than  the  pupil's  reason.  A  method  of  presentation  is  sought  which  will 
be  its  own  explanation,  to  be  comprehended  today  or  next  year. 

Inspection — Much  use  is  made  of  inspection  work.  It  is  no  longer  necessary 
to  find  factors,  divisors,  and  multiples  of  large  numbers,  and  for  this  reason  me- 
chanical methods  of  finding  such  factors,  etc.,  are  discarded,  and  in  their  stead 
are  given  methods  that  will  reinforce  the  power  of  inspection.  Thruout  the 
course  such  methods  are  used  as  will  contribute  to  the  mastery  of  number. 

When  possible  a  topic,  e.  g.,  common  divisors,  is  introduced  for  the  first  time 
in  connection  with  its  use,  and  is  then  dealt  with  only  sufficiently  to  serve  the  pur- 
pose then  at  hand.    It  is  later  taken  up  by  itself  and  given  full  statement. 

Efficiency — That  method  of  teaching  is  most  efficient  which  accomplishes  the 
desired  result  with  the  best  expenditure  of  time  and  eflfort  on  the  part  of  the 
pupil  and  teacher,  without  at  the  same  time  doing  violence  to  the  child's  intelli- 
g-ence  or  will.  Arithmetic  may  be  learned  as  an  assigned  task  accompanied  by 
the  requisite  amount  of  drill,  but  such  a  method  will  produce  mechanical  work- 
ers and  haters  of  arithmetic.  Time  may  be  wasted  on  useless  games  and  exer- 
cises which  do  not  contribute  to  real  advancement.  Both  extremes  should  be 
avoided.  Interesting  plays,  garriies  and  exercises  which  count  in  the  final  equip- 
ment may  be  found.  Such  have  been  sought  in  the  following  pages.  No  game 
has  been  recommended  merely  because  it  is  pleasing,  and  the  exercises  being  in- 
telligent and  purposeful  appeal  to  the  child's  interested  effort.  Crutches  should 
be  avoided,  and  forms  for  workj  and  ways  of  doing  it  which  will  later  be  discard- 
ed should  not  be  introduced.  An  exception  to  the  last  staterment  is  permissible 
when  the  form  finally  adopted  is  an  abbreviation  of  the  fuller  form  which  should 
be  used  at  first :  e.  g.,  the  full  and  the  abbreviated  forms  for  addition  and  sub- 
traction of  fractions. 


X  ARITHMETIC 

Efficiency  demands  that  the  operations  be  performed  in  the  most  direct  man- 
ner, the  method  which  leads  to  the  mastery  of  number  being  preferred.  Inspec- 
tion work  should  be  encouraged  and  labor  saving  devices  should  be  taught  in 
such  a  manner  as  to  challenge  the  pupil's  best  efforts.  Actual  business  forms  and 
business  customs  should  be  taught  and  followed. 

Thoroness  should  be  the  final  goal.  The  processes  that  need  to  be  learned 
are  not  so  numerous  that  they  cannot  be  mastered,  and  the  feeling  of  security  is 
very  satisfying.  There  are  too  many  boys  and  girls  who  are  afraid  when  they 
encounter  a  problem  involving  fractions,  and  who  surrender  when  the  operation 
involves  the  handling  of  a  complex  decimal. 

In  most  business  operations  it  is  true  that  only  small  fractions  are  used,  and 
that  the  business  man  is  satisfied  if  the  final  result  is  correct  to  two  or  three  deci- 
mal places.  It  is  also  true  that  science  and  business  are  seeking  greater  accuracy 
m  small  things.  The  astronomer  measures  time  and  angles  to  a  hundredth  of  a 
second,  the  axle  of  the  automobile  is  measured  to  a  thousandth  of  an  inch,  and 
the  price  of  electricity  is  quoted  in  hundred  thousandths  of  a  cent.  "Now  a  mer- 
chant needs  astronomy  to  see  them  (the  profits),  and  when  he  locates  them, they 
&re  out  some  where  near  the  fifth  decim.al  place".  The  final  result  can  be  ac- 
curate to  two  decimal  places  only  when  the  successive  steps  have  been  kept  well 
in  hand.  A  result  may  be  more  accurate  or  less  accurate  than  the  data  according 
as  the  error  has  been  multiplied  or  divided.  The  pupil  should  be  trained  to  dis. 
criminate. 

It  will  be  seen  that  the  aim  of  all  our  work  is  to  present  the  subject  mat- 
ter in  such  a  way  as  to  meet  the  psychological  conditions  of  the  learners.  As 
the  natural  aptitudes  and  the  environmental  conditions  of  individual  children 
vary  greatly,  it  is  evident  that  in  order  to  reach  efficiency  there  must  be  a 
sectioning  of  the  grades  into  groups,  by  which  the  quick  and  successful  chil- 
dren are  given  freedom  to  advance  at  their  own  rate  while  the  slow  or  unsuc- 
cessful pupils  are  allowed  to  carry  on  their  work  in  the  way  natural  to  them. 
It  is  also  important  that  no  thought  of  inferiority  or  superiority  in  the  work  of 
any  of  the  groups  should  be  given  to  the  children  or  held  by  their  teachers. 
There  can  be  no  efficient  instruction  unless  the  inherent  powers  and  natural 
rhythm  of  the  children  are  considered  in  the  teaching  effort.  The  working 
plans  of  the  Training  School  are  such  as  to  secure  this  division  of  the  grades 
into  small  sections  at  varying  stages  of  progress.  This  makes  possible  that 
adaptation  of  the  work  to  individual  needs  and  powers  without  which  the  most 
careful  methods  of  presentation   do  not  insure  success. 

This  idea  of  definitely  and  frankly  adjusting  the  work  to  the  individual 
appears  again  and  again  in  the  early  pages  of  the  manual  and  is  assumed  thru- 
out  the  book. 


AN  OPEN  LETTER 

From    a    Supervisor    of   Arithmetic   to   the   Students   Teaching 

Arithmetic  in 

THE  TRAINING  DEPARTMENT 

of 

THE  STATE  NORMAL  SCHOOL  AT  SAN  JOSE,  CAL. 


Copyright,    1913,   By  Adelia   R.    Hornbrook. 


Dear  Friends: 

Altho  not  all  of  you  can  be  so  fortunate  as  to  be  assigned  to  the  teach- 
ing of  the  little  children  who  begin  number  work,  you  will  see  at  once  that  it  is 
important  that  every  one  of  you  shall  have  a  clear  understanding  of  the  aims, 
the  principles  and  the  processes  of  these  beginnings  as  well  as  those  of  the  later 
work.  This  is  necessary  in  order  that  the  work  of  each  of  you  during  your  twelve 
weeks  of  practice  may  be  rightly  related  to  that  of  others. 

In  these  few  weeks  you  are  to  begin  your  professional  study  of  children's 
minds  in  their  reactions  upon  the  truths  of  mathematics.  Your  success  in  teach- 
ing will  depend:  first,  upon  your  habit  of  close,  intelligent  observation  of  these 
reactions ;  second,  upon  your  skill  in  interpreting  what  you  observe ;  and  third, 
upon  your  ability  to  present  the  facts  of  mathematics  in  ways  suitable  to  the 
minds  of  the  children  as  you  find  them  to  be.  This  letter  is  written  to  help  you. 
It  refers  to  the  work  of  the  first  four  grades. 

There  are  two  ideas  of  number,  viewed  in  the  light  of  child  psychology, 
which  underlie  the  plans  of  teaching  here  presented.  These  are  the 
"number  series  idea"  and  the  "number  form  idea."  If  you  are  to 
use  the  plans  intelligently  you  will  need  to  understand  these 
basic  principles.  In  Dr.  Stanley  Hall's  Educational  Problems,  Vol.  2,  pp.  350- 
356,  you  v.'ill  find  an  excellent  statement  of  the  Series  Idea  in  Number,  and  many 
known  facts  about  Number  Forms,  with  a  short,  encouraging  reference  to  the 
application  of  these  ideas  in  the  practical  plans,  some  of  which  are  given  in  this 
letter. 

The  Series  Idea  The  number  series  idea  was  brought  to  the  notice  of  American 
In  Number  educators  by  an  article  in  the  Pedagogical  Seminary  for  October, 
1897,  written  by  Dr.  D.  E.  Phillips,  then  of  Clark  University.  If 
is  now  generally  accepted  by  writers  on  primary  arithmetic  and  is  practically 
applied  in  most  modern  textbooks  of  that  subject. 

Very  briefly  stated,  the  idea  is  this ; — 

Ordinary  children  in  their  early  years  think  of  numbers  as  a  series  of  sounds, 
"one,  two,  three."  etc.  They  like  to  bring  this  series  into  their  consciousness  and 
play  with  it.  They  repeat  these  number  words,  generally  attaching  no  more 
meaning  to  them  than  to  "eeny,  meeny,  miny  mo,"  but  enjoying  the  rhythms,  the 
repetitions,  and  the  jingle  of  this  sound  series.  They  love  to  show  to  grown-ups 
their  new  and  interesting  accomplishment  of  counting. 

Your  first  work  will  be  to  find  out  how  well  your  individual  children  can 
count  and  then  to  help  them  to  count  perfectly  to  100.  This  is  as  pleasant  to  them 
as  any  play.  There  may  be  a  little  stiffness  or  shyness  at  first,  but  that  will  dis- 
appear as  you  get  into  happy,  sympathetic  relations  with  the  little  people. 


2  ARITHMETIC 

Query.  Can  you  recall  any  instances  in  your  school  life  in  which  your  suc- 
cess in  learning  was  affected,  either  favorably  or  unfavorably  by  your  feelings 
toward  your  teacher  or  hers  toward  you? 

Note.  The  queries  scattered  thru  this  letter  are  to  be  answered  by  you 
at  the  weekly  conferences. 

Number  The  discovery  of  the  existence  in  many  minds  of  certain  definite 

Forms  visualizations  called  "number  forms"  was  given  to  the  world  in  1883 
by  Sir  Francis  Galton,  in  the  book,  Inquiry  into  Human  Faculty. 
It  has  been  confirmed  by  many  later  investigators. 

Stated  in  very  condensed  form,  the  facts  are  these : — 

Many  children  in  their  early  gropings  among  numbers  and  figures,  make  a 
mental  picture  of  the  number  series,  usually  up  to  100,  sometimes  far  beyond. 
They  visualize  the  number  symbols,  i,  2,  3,  etc.  as  a  succession  of  figures.  They 
see  them  mentally  at  definite  distances  and  directions  from  one  another  as  if  on 
a  printed  page.  The  lines  of  figures  thus  formed  in  the  mental  picture  are  some- 
times straight  or  broken,  sometimes  curved,  sometimes  in  spirals,  sometimes 
in  parallels.  They  differ  in  different  minds.  It  is  estimated  by  psychologists  that 
about  five  per  cent  of  adults  retain  and  use  the  number  forms  that  they  built  up 
in  childhood,  as  a  help  in  working  with  numbers.  Sometimes  the  forms  are  very 
complicated,  twisted  and  irregular  like  one  that  was  given  to  me  last  year  by  a 
Junior  student.  Altho  it  seemed  a  very  inconvenient  form  to  use,  she  assured  me 
that  it  was  a  constant  help  to  her  in  reckoning.  Of  the  many  hundreds  of  num- 
ber forms  that  have  been  reported,  mostly  irregular,  only  one  was  complained  of 
by  its  possessor  as  being  "troublesome"  on  account  of  the  bending  and  doubling 
of  its  lines.  But  certainly  the  possessor  of  an  even,  regular  number  form  like 
that  given  to  me  by  a  teacher  in  our  school  last  year  is  fortunate.  You  will  find 
in  the  school  library  a  most  interesting  and  instructive  article  upon  this  subject. — 
The  Genesis  of  Number  Forms, — by  Dr.  D.  E.  Phillips,  in  the  American  Jour- 
nal of  Psychology,  Vol.  8,  No.  4. 

The  facts  concerning  these  spontaneous  visualizations  of  the  number  series 
have,  as  Pres.  Butler  of  Columbia  University  remarked  editorially  (Educational 
Review,  May,  1893),  "a  most  direct  bearing  upon  the  teaching  of  elementary 
arithmetic."  They  are  of  great  practical  importance  to  us,  for  instead  of  allowing 
our  pupils  to  form  irregular,  inconvenient  mental  diagrams  of  the  number  series, 
or  none  at  all,  we  are  going  to  give  to  each  child  by  means  of  charts  and  other 
apparatus  an  opportunity  to  use  freely  a  visible,  tangible  representation  of  the 
series  up  to  too  in  a  regular  unchanging  form.  With  this  he  can  make  his  own 
discoveries  of  the  facts  of  number,  or  can  readily  perceive  the  number  facts 
pointed  out  by  his  teachers  and  classmates. 

This  is  not  a  new,  untried  project.  Plans  based  upon  this  idea  I  began  to 
work  out  in  1886,  presenting  some  of  them  in  an  educational  magazine  in 
1893  and  in  a  textbook  in  1898.  For  many  years  I  have  had  the  pleasure  of 
kno'wing  that  other  educators  were  working  with  the  same  thought.  In  the 
recent  writings  of  some  leading  California  educators  the  use  of  charts  similar 
to  ours  is  urged.  One  of  them  is  given  in  our  state  textbook  of  primary  arith- 
metic (p.  17).  Some  happy  results  were  obtained  in  our  school  last  year  by  the 
student  teachers  of  one  grade  who  used  some  of  these  plans  quite  successfully  with 
a  short  period  of  supervisory  help.  We  are  expecting  fine  results  from  your  work 
this  year,  and  we  are  planning  to  give  you  all  the  supei^visory,  informational 
help  )'-ou  need, — and  not  a  bit  more.  We  want  you  to  be  independent,  alert, 
fertile-minded  workers. 

This  year  (1913)  there  will  be  three  or  four  grades  beginning  number  work 
from  the  3  B  down.  The  work  here  described  is  planned  for  first  and  second 
grade  pupils,  and  is  made  suitable  for  children  thru  the  3  B  grade  by  simply  al- 
lowing them  to  advance  more  rapidly. 


OPEN  LETTER 


The  First 
Lessons 

In  the  first  lessoi 
before  the  class. 

IS  the  ni 

umber  cl 

lart  of  f 

ives,  as 

given  be 

low  * 

I 

11 

21 

31 

41 

51 

61 

71 

81 

91 

2 

12 

22 

32 

42 

52 

62 

72 

82 

92 

3 

13 

23 

33 

43 

53 

63 

73 

83 

93 

4 

14 

24 

34 

44 

54 

64 

74 

84 

94 

5  15    25    35    45    55    65    75    85    95 

6  16  26    36  46  56  66  76  86    96 

7  17  27    37  47  57  67  77  87    97 

8  18  28  38  48  58  68  78  88    98 

9  19'  29  39  49  59  69  79  89        99 
10  20  30  40  50  60  70  80  90  100 

The  chart  is  copied  on  the  board  in  large,  plain  figures.  The  multiples  of 
five  are  written  in  larger  figures  than  the  other  numbers  with  crayon  of  some 
light  color,  never  in  a  dull  color. 

Queries.  Why  should  these  figures  be  large  and  bright?  Can  you  give  any 
psychological  fact  that  suggests  a  reason? 

There  are  three  different  things  which  the  child  must  master  in  his  early 
work.    They  must  not  be  confused.    They  are : 

1st,     The  sound  series  up  to  100. 

2nd.    The  sight  series  up  to  100. 

3rd.    The  use  of  these  series  in  applying  number  to  objects. 

It  will  take  many  weeks  for  the  little  ones  to  master  these  three  basic 
elements.  For  those  below  the  3  B  grade  fifteen  or  twenty  minutes  a  day  are  al- 
lowed; for  the  3  B  and  those  above,  forty  minutes.  Each  group  will  advance  at 
its  own  rate,  doing  what  it  can  from  day  to  day  without  hurry  or  worry.  We 
shall  use  many  different  exercises  and  simple  childish  plays.  Children  naturally 
love  number  and  they  love  to  play.  We  will  combine  these  two  natural  interests 
and  as  a  result  will  obtain  clear  perceptions  of  number,  and  the  joy  which  chil- 
dren find  in  doing  something  worth  while  in  the  company  of  their  mates. 

Organized,  purposeful  play,  guided  by  the  teacher,  leads  to  voluntary,  result- 
ful  work  by  the  pupils. 

*The  charts  in  this  pamphlet  are  taken   from  Hornbrook's   Primary  Arithmetic  by 
permission  of  the  American  Book  Co. 


4  ARITHMETIC 

Learning  The  Most  children  beginning  number  work  know  a  part  of  the 

Sound  Series  sound  series.     They  can  count  a  little  way  with  more  or  less 

correctness.     So  we  shall  begin  by  using  this  knowledge  as  a  basis, 
passing  gradually  "from  the  known  to  the  unknown." 

There  are  many  good  ways  of  beginning  the  study  of  number.  I  have  selected 
one  of  them  and  will  describe  some  typical  work  of  a  teacher,  whom  we  will  call 
Miss  Smith  and  will  image  as  a  young  woman  of  charming  personality,  working 
with  one  of  our  Training  School  groups  of  about  a  dozen  children.  This  work 
is  not  given  as  a  model  to  be  copied,  but  as  a  series  of  concrete  illustrations  of 
principles  to  be  interpreted.  Thoughtfully  read,  it  will  give  you  mental  pictures 
of  schoolroom  activities  differing  probably  from  those  in  which  you  figured  as 
a  child.  Educational  thought  has  changed  very  rapidly  in  the  last  few  years,  and 
you  have  come  to  the  San  Jose  Normal  to  get  the  thought  of  the  present,  not  of 
the  past.  Your  previous  training  in  the  department  of  Psychology,  reinforced  by 
special  reading,  will  enable  you  to  understand  the  psychological  principles  upon 
which  the  plans  are  based. 

It  is  the  first  lesson.  Miss  Smith  begins  in  the  usual  way  by  talking  with  the  children 
about  counting.  One  child  thinks  he  can  count  to  30,  another  believes  that  he  can  count 
"a  whole  lot"  and  so  on.  "Now  all  count  with  me,'"'  she  says,  and  begins  to  count  slowly. 
The  "counting"  is  simply  giving  the  number  names  in  their  true  order.  Their  significance 
is  not  considered  at  all.  That  comes  later.  The  children  are  now  getting  the  sounds  in 
their  true  sequence.  As  the  counting  goes  on,  the  teacher  makes  mental  notes  of  the  pupils 
who  fall  by  the  wayside,  and  of  those  who  go  on  triumphantly  until  she  gives  the  signal 
to  stop.    The  signal  is  given  as  soon  as  she  sees  signs  of  failing  powers  or  flagging  interest. 

In  this  first  lesson  the  young  teacher  begins  her  study  of  the  minds  of  the  children 
whom  she  is  to  teach.  "What  can  this  child  do  and  how  does  he  do  it,  and  how  can  I  best 
help  him  to  do  the  next  thing?"  are  frequently  recurring  thoughts. 

In  the  ideal  conditions  of  your  work, — with  your  groups  of  about  a  dozen 
children,  each  in  its  own  pleasant  little  classroom,  with  sympathetic  supervisory 
help  close  at  hand  and  with  the  children  looking  to  you  as  the  bringer  of  some- 
thing new  and  interesting,  it  will  be  only  a  short  time  until,  if  you  use  intelligent 
effort,  you  Avill  be  able  to  see  your  whole  group  as  individuals,  each  with  his  own 
abilities  and  temporary  inabilities,  which  you  are  to  help  him  remove.  Then  the 
true  interchange  of  thought  between  teacher  and  pupil  will  take  place.  You  will 
watch  and  assist  the  development  of  the  mathematical  sense  of  each  child.  He 
will  learn  to  look  upon  you  as  his  own  personal  helper.  The  more  clearly  he  sees 
you  in  that  light,  the  more  trustfully  and  happily  he  will  follow  your  guidance. 
When  this  beautiful  intimacy  of  thought  is  established,  the  need  for  discipline — 
in  the  old  harsh  sense — disappears.  Instead  we  shall  have  obedience,  pleasure, 
and  clear  understanding. 

Because  the  presence  of  outsiders  breaks  the  flow  of  thought  between  pupils 
and  teachers,  you  and  your  children  are  to  be  shielded  as  much  as  possible  from 
outside  visitors.  The  parents  of  the  pupils  are  more  than  welcome.  We  are 
anxious  to  consult  them.  They  can  help  us  to  understand  the  children.  It  will 
be  well  to  have  it  understood  that  when,  for  instance,  Mary's  mother  or  father 
comes,  Mary  shall  do  an  unusual  amount  of  reciting  in  order  that  they  may  judge 
of  her  attainments 'or  defficiencies.  And  so  with  each  child.  Parents  who  have  en- 
trusted their  children  to  our  care  are  in  sympathy  with  us  in  our  work.  They 
have  a  right  to  know  of  their  children's  success  and  of  their  temporary  tailures, 
if  they  care  to  follow  the  course  of  instruction  closely.  Altho  we  earnestly  desire 
that  they  shall  not  break  that  course  by  attempting  to  teach  their  children,  we 
are  anxious  to  learn  from  them  all  we  can  about  the  ways  in  which  the  minds  of 
our  pupils  react  upon  the  work. 

Let  us  return  to  our  ideal  teacher  whom  we  left  watching  her  little  ones  as 
they  counted. 


OPEN  LETTER 


After  the  group  counting,  which  has  served  to  break  the  ice,  the  individual  activities 
begin,  the  voluntary  work  or  "stunts"  as  they  are  called.  "Who  wants  to  count  all  alone 
for  us?"  says  Miss  Smith.  Volunteers  are  ready  with  the  usual  signal.  Tom  is  chosen.  Glad 
of  his  chance  to  display  his  accomplishments,  he  begins  to  rattle  off  the  numbers. 

A  point  of  importance  suggests  itself  here. 

Perhaps  Tom  blunders,  puts  19  directly  after  15.  As  soon  as  the  mistake  is 
corrected  either  by  himself  or  by  some  one  designated  by  the  teacher,  he  must 
stop.  Perfection  is  the  standard  and  nothing  short  of  it  is  acceptable.  The  slight- 
est error  "spoils  the  stunt."  This  rule  is  to  be  invariable  and  is  explained  at  the 
first  lesson  in  an  easy,  pleasant  way  as  "the  way  we  play  the  game."  So  Tom 
sits  down  encouraged  by  a  word  or  a  look  from  the  teacher  to  feel  that  he  will 
"soon  do  it  all  right."  Mary  has  the  floor,  and  Tom,  instead  of  stumbling  along  in 
a  maze  of  uncertainty  with  half  grasped,  embarrassing  corrections  from  his 
teacher,  hears  ISIary's  quiet  smooth  counting.  If  he  is  an  alert,  sensitive  child, 
he  follows  Mary's  performance  attentively  with  a  keen  desire  to  equal  it.  If  he 
is  slow,  unawakened  as  yet,  his  attention  may  be  slack,  but  he  is  unconsciously 
becoming  familiarised  zvith  the  num,ber  series. 

Another  point.  Henry  may  say  with  pride,  "I  can  count  to  100,  and  Tom 
can  only  count  to  15."  This  is  Henry's  mistaken  way  of  trying  to  win  the  ap- 
proval of  his  teacher  and  classmates.  It  is  contrary  to  the  spirit  of  respect  and 
consideration  for  others  which  is  to  rule.  The  teacher  may  say,  "Can  you?" 
adding  in  a  low,  confidential  tone  with  a  very  significant  look,  "But  you  mustn't 
say  anything  about  it."  Or  she  may  answer  in  a  light,  easy  way,  "Well,  if  that 
is  so  we  will  have  to  help  him,  won't  we?" 

Query.     To  -what  feeling  does  each  of  these  answers  appeal? 

The  oral  counting  is  continued  from  day  to  day,  along  with  the  chart  work  and  the 
counting  of  objects  which  I  shall  describe  later.  After  a  time  Miss  Smith  introduces  decade 
counting.  "We  will  count  by  'parts'  now,"  she  says.  "Who  can  begin  at  21  and  count?" 
Later  as  a  little  anticipation  of  the  process  of  adding  numbers  to  20,  30,  etc.,  she  asks,  "Who 
can  begin  at  41  and  count  five,"  or  "Who  is  ready  to  begin  at  71  and  count  ten?"  Not 
quite  knowing  her  pupils  yet,  she  sometimes  suggests  feats  of  ability  that  prove  to  be  be- 
yond them.  In  that  case  she  quickly  and  smilingly  substitutes  something  else.  She  does 
not  invite  failure  and  confused,  unhappy  thinking  by  forcing  upon  her  pupils  anything  for 
which  she  sees  that  they  are  not  yet  ready. 

One  of  the  wisest  things  written  lately  about  education  is  the  following  from 
a  little  book; — The  Montessori  System — by  Dr.  Theodate  Smith,  of  Clark  Uni- 
versity. "Failure  is  a  negation  showing  that  the  child  is  not  yet  ready  for  that  par- 
ticular exercise,"  p.  vii. 

A  child  may  advantageously  see  work  done  by  his  mates  that  he  does  not 
yet  understand.  He  will  not  be  confused  by  it  unless  he  himself  is  called  upon 
for  it.  Hence  the  value  of  the  principle  of  "voluntary  effort"  used  so  much  in 
our  school  and  in  other  schools  of  the  modern  type.  Calling  for  voluntary  work 
the  teacher  is  careful  to  see  that  each  child  has  the  opportunity  to  express  his 
thought,  and  she  cares  particularly  for  the  weak  little  ones  who  need  her  help 
before  they  are  able  to  present  the  results  of  straight  thinking. 

It  is  to  be  observed  that,  altho  much  freedom  is  allowed.  Miss  Smith's  class- 
room is  not  ruled  by  the  children's  whims.  She  knows  that  she  is  wiser  than  her 
pupils  and  that  upon  her  rests  the  responsibility  of  the  work,  that  the  children  are 
there  to  learn  and  that  they  must  not  be  hindered  by  the  chaotic  conditions  which 
would  arise  without  a  competent  person  in  control.  She  is  the  gentle,  calm, 
confident  director  of  affairs  whose  directions  are  to  be  obeyed.  That  is  what  we 
expect  you  to  be  in  your  classrooms.  The  principal  of  the  Training  School  and 
every  one  of  your  supervisors  will  give  you  all  needed  support  in  that  position. 


ARITHMETIC 


Learning  The  At  first  the  children  see  the  chart  as  a  mixed-up  blurry  pic- 

Sight  Series  ture  of  meaningless  lines  and  surfaces.  But  the  series  of  number 

words  which  they  have  learned  is  the  key  that  will  unlock  for  them  all  its  values. 

After  the  oral  counting  in  the  first  day's  lesson,  Miss  Smith  begins  the  chart  work 
that  is  to  lead  to  the  knowledge  cf  the  sight  series,  and  to  the  realization  of  many  of  the 
relations  of  numbers.  Calling  attention  to  the  chart  she  says,  "All  the  numbers  that  you 
have  been  saying  are  on  this  chart,  and  I  am  going  to  show  you  how  to  find  them."  (With 
her  little  ones  she  does  not  make  the  distinction  between  "numbers"  and  "figures"  that  exact 
mathematical  phrasing  would  demand,  nor  can  we.)  She  begins  to  name  in  order  very 
deliberately  the  numbers  in  the  first  column,  pointing  to  each  as  she  names  it.  She  soon 
stops  and  calls  on  the  group  to  count  the  numbers  while  she  points.  Then  she  asks,  "Who 
can  point  while  we  count?"  She  chooses  Louise,  whose  power  of  attention  is  weak.  The 
child  is  glad  to  take  the  pointer,  but  she  is  unable  to  keep  in  touch  with  the  counting  of 
the  group,  and  soon  has  to  turn  the  pointer  over  to  some  one  else.  Miss  Smith  does  not 
show  any  sign  of  disapproval  either  by  word  or  look,  nor  does  she  feel  any  disapproval.  She 
makes  it  easy  for  the  child,  just  as  she  would  for  a  guest.  This  is  the  custom  in  her 
classroom. 

The  ideals  of  courtesy,  when  realized  in  the  schoolroom,  prevent  children 
from  expressing  contempt  or  other  unpleasant  feelings  toward  their  mates.  And 
in  this  training  school  of  ours,  in  which  you  and  I  have  the  sacred  responsibility 
of  influencing  the  dawning  thoughts  of  children  and  in  which  this  described  work 
of  our  imagined  Miss  Smith  is  really  the  suggested,  prophetic  outline  of  your  own 
work,  there  are  not  only  ideals  of  courtesy  and  good  form  but  there  are  ideals 
still  higher.  We  have  all  heard  them  beautifully  expressed  by  the  principal  of  the 
school  and  by  other  teachers — the  ideals  of  love  and  mutual  helpfulness  and  so- 
cial service.  These  ideals  are  to  be  found  not  only  in  our  Normal  School  but  in 
many  schools  thruout  the  country.  They  are  a  part  of  the  best  educational  think- 
ing of  the  present  day.  They  are  displacing  the  selfish,  individualistic  thought  of 
the  old  rod-ruled  schools. 

Miss  Smith  is  trying  to  hold  up  these  higher  ideals,  trying  to  do  it  in  a  quiet, 
non-preaching  way  by  her  own  attitude  towards  the  failing  ones,  and  by  her  de- 
cisions in  the  little  social  questions  that  arise  in  the  classroom.  As  far  as  they 
are  realized  these  nobler  ideals  prevent  the  children  not  only  from  expressing, 
but  from  having  unpleasant  thoughts  about  their  neighbors. 

In  the  case  of  Louise  the  teacher  knows  that  blame,  reinforced  by  the  scorn 
of  the  class,  would  be  positively  harmful  to  the  child,  and  so  she  protects  her 
Louise  can  see  that  she  has  failed  in  this  matter.  Sooner  or  later,  unless  she  is 
subnormal,  the  desire  to  equal  her  mates  and  the  interest  in  learning  will 
lead  her  to  give  voluntary  attention.  In  the  meantime  while  the  power  of  at- 
tention is  strengthening.  Miss  Smith  tries  to  be  helpful  by  giving  the  child  fre- 
quent opportunities  to  take  part  in  the  exercises  under  favorable  conditions. 

In  this  description  you  will  observe  that  I  do  not  fix  the  limit  of  a  day's  les- 
son, but  simply  give  the  exercises  in  a  continuous  story,  in  the  order  in  which  they 
follow  one  another.  Of  course  only  one  new  exercise  should  be  given  at  a  lesson. 
The  amount  of  work  and  the  number  of  different  exercises  to  be  used  each  day, 
depend  upon  the  way  in  which  the  children  respond  to  the  work  and  upon  the 
teacher's  judgment  of  it  at  the  time.  No  grade  limits  are  given  here.  Each 
student  teacher  will  take  up  the  work  at  the  beginning  of  a  term  where  her  pred- 
ecessor has  left  it. 

Miss  Smith  starts  the  "stimts"  in  the  learning  of  the  sight  series  by  asking,  "Who 
wants  to  count  and  point  all  alone?"  George  succeeds  in  counting  to  the  end  of  the 
column.  "Stop  at  the  10,"  says  the  teacher.  None  are  allowed  to  go  beyond  10  that  day 
nor  for  many  days. 

Query.  Why  should  the  pupils  not  be  allowed  to  go  beyond  the  first  ten 
numbers  at  this  time? 

A  person  entering  the  room  might  suppose  that  the  children  were  reading  the  figures, 
but  generally  they  are  as  yet  only  finding  the  names  of  the  figures  by  counting.  As  the 
next  step  Miss  Smith  points  to  the  figure  3  and  turning  to  one  of  her  quickest  pupils  says : 
"Ruth,  what  number  is  this?"  "3,"  replies  Ruth.  "How  did  you  find  out?"  Ruth  takes  the 
pointer  and  explains,  "I  began  up  here  with  the  1,  and  I  just  counted  right  down  and  when 


OPEN  LETTER  1 

I  said  3  that  was  it."  Thus  taught  and  helped  by  reinforcing  suggestions  from  the  teacher, 
others  name  figures.  It  is  explained  to  them  that  they  must  take  all  the  time  they  need, 
and  be  very  careful.  If  they  hurry  or  are  careless  they  may  make  a  mistake,  which  would 
be  a  very  bad  thing  indeed. 

On  another  day  after  they  have  "named  numbers"  for  a  short  time  they  begin  to 
"hunt  numbers." 

In  the  previous  exercise  a  figure  was  pointed  out  and  the  pupils  had  to  give 
its  name.  In  this  "hunting"  game  the  name  is  given  and  the  child  must  find  the 
figure.  These  two  reverse  ways  of  relating  the  number  symbol  and  its  name  are 
used  by  Miss  Smith  in  all  the  little,  childish  plays  by  means  of  which  she  leads  to 
the  easy,  accurate  reading  of  numbers.  These  forms  of  play,  like  the  oral  count- 
ing and  the  chart  counting,  are  given  on  successive  days  until  no  longer  needed. 

"Look  at  the  chart  and  find  4."  says  Miss  Smith.  They  look  as  directed  and  count 
silently,  some  making  gestures  towards  the  chart  until  they  come  to  "4,"  and  are  ready 
to  point  out  the  figure.  Unwilling  that  the  class  shall  hear  an  error,  the  teacher  is  careful 
to  call  at  first  upon  the  most  apt  pupils.  When  the  exercise  is  understood  she  gives  a  new 
form  to  it  by  asking,  "Who  knows  a  number  for  the  class  to  find?"  Helen's  hand  is  raised. 
"Come  and  whisper  it  to  me,"  says  Miss  Smith.  Helen  comes  forward  and  whispers,  "5," 
"Give  it  to  the  class,"  is  the  direction.  When  it  has  been  found  Helen  looks  shyly  up  at 
Miss  Smith  and  says,  "I  know  where  another  number  is."  She  is  allowed  to  give  it  to  the 
class  and  they  find  it. 

A  child  who  comes  forward  and  directs  the  group  work  is  called  a  "leader." 
Nearly  every  child  loves  to  "play  leader."  Of  course  a  blunder  robs  him  of  that 
dignity  at  once,  but  it  is  often  necessary  for  a  pupil  to  give  up  leadership  wriile 
his  work  is  still  perfect  in  order  to  give  the  others  a  chance.  These  are  called 
"true  leaders."  The  teacher  keeps  a  list  of  the  true  leaders.  Later  they  will  be 
allowed  to  do  special  work. 

At  this  stage  the  children  generally  are  not  yet  able  to  recognize  the  figures  at  sight. 
They  are  learning  to  see  them  as  separate  things  in  the  chart  picture  and  the  next  step  is 
to  learn  their  forms.  Miss  Smith  has  a  set  of  figures,  six  or  seven  inches  high,  mounted 
on  pasteboard.  (This  September  I  am  going  to  lend  you  a  set  until  you  can  get  them  from 
ralendars).  She  sits  before  the  class,  holds  up  the  1  and  running  her  finger  down  its  length, 
says,  "Who  wants  to  come  up  and  go  over  this  nice  big  1  just  as  I  do?"  Every  one  except 
a  very  slow,  shy  little  fellow  is  eager  to  come.  After  several  have  "gone  over"  the  big  1, 
moving  a  little  bunch  of  fingers  over  it  without  touching  it,  she  turns  to  the  shy  child  and 
in  an  easy,  off-hand  way  calls  upon  him  to  come  and  try.  If  he  still  shrinks  she  excuses  him 
for  the  present,  saying  to  herself,  "I  must  look  out  for  that  little  chap.  He  will  need  special 
attention." 

In  our  school  where  the  principle  of  voluntary  effort  is  applied  in  so  many 
delightful  ways,  some  of  which  you  can  see  any  day  by  spending  the  assembly 
period  in  one  of  the  school  halls  of  assembly,  it  is  unusual  to  find  a  child  who 
would  refuse  such  an  opportunity  as  Miss  Smith  offers.  Such  a  child  needs  to 
have  his  work  skillfully  adapted  to  his  capacity.  Gradually  under  the  sunshine 
of  the  teacher's  kind  thought  in  the  genial  atmosphere  of  a  classroom  in  which  con- 
sideration for  others  is  the  ideal,  the  little  one's  fear  and  reserve  will  melt  away. 
He  will  begin  to  work.  And  then  how  happy  he  will  be  when  he  finds  himself 
doing  the  things  that  the  others  are  doing,  those  puzzling,  embarrassing  things 
that  he  had  thought  too  hard  for  him  even  to  try. 

The  big  1  having  been  traced,  Miss  Smith  holds  up  the  2.  This  is  more  difficult.  Some 
start  at  the  wrong  place  or  go  in  the  wrong  way.  The  small  hands  have  to  be  guided.  "You 
must  always  begin  at  the  nose  of  the  2,"  said  a  little  girl,  "and  go  right  round  to  the  tail; 
and  you  mustn't  ever  rub  the  fur  the  wrong  way." 

In  the  comments  which  our  pupils  are  encouraged  to  make,  subject  to  the 
laws  of  courtesy  and  good  sense,  they  show  to  us  the  workings  of  their  minds 
and  they  often  give  useful  suggestions.  A  little  boy  in  one  of  my  experimental 
classes  last  year  called  the  big  2,  Avhich  I  gave  him  for  a  copy,  a  "papa-2,"  and 
proceeded  to  make  a  smaller  one  which  he  called  a  "mamma-2,"  and  one  still 
smaller  which  he  called  a  "little-2,"  This  interested  other  pupils  and  one  of  them 
added  a  tiny  little  figure  which  he  called  a  "baby-2."  We  allow  the  pupils  to  ex- 
press their  childish  fancies  but  do  not  impose  them  upon  other  children  to  whom 
they  do  not  appeal. 

Miss  Smith  does  not  introduce  any  more  figures  that  day  because  she  is  not  willing 
to  risk  confusing  her  pupils  by  presenting  too  many  forms  at  one  lesson. 


8  ARITHMETIC 

Queries.  When  several  strangers  are  introduced  to  you  at  one  time  do  you 
ever  have  difficulty  in  applying  the  right  names  to  them  afterwards?  You  have 
an  adult  brain. .  How  about  the  brains  of  little  children  in  comparison  with  yours? 
Tomorrow  Miss  Smith  will  have  big  2's  written  on  the  board,  one  for  each  child  to 
trace.  Those  who  do  it  right  will  be  allowed  to  take  a  crayon  and  trace  the  figure.  Then 
they  will  copy  it.  Arm  movement  "with  a  good  big  swing"  is  the  ideal  she  will  set  before 
them.  When  2  is  so  well  learned  that  it  can  be  recognized  anywhere  she  will  present  the 
3  in  the  same  way,  then  the  other  digits  in  their  order.  She  never  presents  more  than  one 
digit  at  a  lesson.  When  4  is  reached,  Miss  Smith  does  not  require  the  pupils  to  trace  the 
printed  figures  which  she  shows.  Instead  she  tells  them,  "When  we  write  the  4  we  always 
leave  the  top  open."  Then  she  sets  them  to  tracing  and  copying  the  large  4's,  that  she  has 
written  on  the  board.  She  knows  that  children  are  apt  to  turn  the  6  the  wrong  way,  so  she 
gives  them  much  practice  on  that  figure,  insisting  upon  the  arm  movement. 

An  interesting  psychological  explanation  of  this  tendency  to  reverse  figures 
and  drawings  has  been  given  and  we  will  discuss  it  at  one  of  our  weekly  confer- 
ences. At  any  rate  we  will  prevent  our  pupils  from  reversing  their  figures  if  we 
can. 

There  are  two  ways  of  tracing  the  8,  Miss  Smith  insists  upon  the  movement  used  by 
the  supervisor  of  writing. 

Each  day  before  taking  up  the  new  figure  the  pupils  write  those  already  learned  in 
a  column  like  the  first  column  of  the  chart.  It  pleases  them  to  see  their  columns  grow  as 
the  new  figures  are  added,  and  their  interest  grows  with  it.  If,  however,  any  member  ot 
the  group  blunders  in  recognizing  or  copying  a  figure  already  given  no  new  figure  will  be 
given  that  day.     Instead  some  other  exercise  will  be  used. 

With  regard  to  this  matter  of  diverting  attention  from  errors  you  will  find 
one  thing  which  may  seem  to  you  at  first  to  be  strange,  altho  experienced  teach- 
ers know  it  well  and  profit  by  it.  When  your  pupils  get  wrong  ideas  and  their 
little  brains  are  "all  muddled  up,"  if  you  labor  with  them  showing  them  the 
right  thing  in  contrast  with  the  wrong,  you  will  only  make  matters  worse.  All 
your  strenuous  efforts  are  worse  than  wasted.  Instead  of  laboring,  the  skilful 
teacher  simply  shows  the  right,  briefly  and  easily,  and  then  turns  to  something 
else.  On  another  day  she  leads  carefully  up  to  that  upon  which  the  pupils  were  con- 
fused. The  chances  are  that  in  the  interval  certain  little  modifications  of  their 
brains  have  taken  place  as  the  result  of  her  work  and  of  the  children's  desire  to 
know  its  meaning  and  that  because  of  this  interval  the  subject  is  easily  made 
clear  to  them.  If  not,  it  is  evident  that  they  are  "not  yet  ready"  for  that  particular 
exercise. 

Query.  Why  is  it  in  the  long  run  a  saving  of  time  and  effort  to  present  the 
digits  so  slowly  and  carefully? 

Many  other  exercises  are  going  on  in  Miss  Smith's  room  from  day  to 
day  by  which  the  children  are  learning  the  sight  series  and  all  unconsciously 
getting  the  first  dim  ideas  of  the  relations  of  numbers,  ideas  that  zvill  brighten 
into  consciousness  later. 

In  one  of  the  first  lessons  Miss  Smith  supplied  each  child  with  a  chart  for  his 
own  personal  use.  Soon  she  will  have  her  pupils  make  the  charts  for  themselves. 
The  chart  was  made  as  follows : 

Upon  a  piece  of  Manila  paper  one  foot  square  (obtained  from  the  supply  rooms)  she 
drew,  very  lightly,  a  ten-inch  square.  She  divided  it  into  inch  squares  by  light,  almost 
invisible  lines,  and  wrote  the  hundred  numbers  in  the  squares.  The  multiples  of  five  are 
larger  than  the  other  numbers  and  are  written  in  bright  colors  with  cra)'ola.  After  the 
children  have  examined  their  charts  and  given  all  the  comments  or  questions  that  she  thinks 
worth  while,  she  sets  them  to  counting  and  pointing  out  the  numbers.  Then  (probably  on 
the  next  day)  she  gives  to  each  child  a  folded  paper  containing  squares  of  pasteboard 
slightly  smaller  than  inch-squares  upon  which  are  pasted  printed  figures  cut  from  calendars 
or  made  by  rubber  stamps.  (Written  figures  may  be  used,  but  they  should  not  be  written 
with  a  pen.  They  need  to  be  clear  and  bold.  If  cotton  is  wrapped  smoothly  around  the 
point  of  a  pencil  and  the  pencil  is  dipped  in  ink,  figures  can  be  made  with  it  almost  as  clear 
and  bright  as  print.)  Multiples  of  five  are  distinguished  by  color  and  size.  Each  paper  con- 
tains only  the  first  ten  numbers.  She  allows  the  children  to  work  in  pairs  if  they  choose. 
The  thing  to  be  done  is  to  place  the  pasteboard  numbers  upon  the  corresponding  chart 
numbers.  To  place  them  where  they  belong  and  "right  side  up  with  care"  is  not  an  easy 
matter  for  the  little  people.  It  furnishes  interesting  occupation  for  many  days,  a  short 
time  each  day. 


OPEN  LETTER  9 

Some  day  after  the  ten  numbers  are  placed  on  the  charts  in  due  order,  Miss  Smith 
shows  the  children  "how  to  play  with  hidden  numbers."  The  first  direction  is,  "Turn  the 
squares  over  just  where  they  are."  "Now  the  numbers  are  all  hidden  out  of  sight,"  she  says. 
"Let  us  see  if  everybody  knows  where  they  are."  Stepping  to  Roy's  desk,  she  puts  a  finger 
on  a  pasteboard  square  and  asks,  "What  is  under  this?"  "6,"  he  replies  "How  did  you 
know,  Roy?"  "I  counted  them.  I  know  them  all.  This  is  4  and  this  is  9  and — "  She  stops 
him  to  address  the  class.  "Be  ready.  Roy  and  I  are  coming  around  to  see  if  you  know 
where  the  numbers  are  hidden."  "What  is  it,  Walter?"  she  says  to  a  boy  whose  hand  is 
raised.  "T  know  them  all,  too."  He  quickly  designates  some  of  the  numbers.  "Walter 
will  take  this  outside  row,  Roy  the  other,  and  I  will  take  the  middle  rows. 

Query.  Why  is  it  better  for  her  to  take  the  middle  rows? 
Another  day  after  the  numbers  are  hidden  she  starts  the  game  of  "letting  the  numb- 
ers out."  "We  are  going  to  let  the  numbers  out  of  the  places  where  they  are  hidden,"  she 
says.  "Uncover  4."  The  children  count,  if  necessary,  until  4  is  reached,  pick  up  the  square 
and  lay  it  over  to  the  left  of  the  chart  column.  The  teacher  calls  on  James  to  "play  leader." 
He  comes  forward  and  designates  the  numbers  to  be  uncovered,  leaving  her  free  to  watch 
individuals.  Some,  without  counting,  know  the  positions  of  the  hidden  numbers.  There 
is  no  chance  of  a  mistake  if  the  child  is  careful  to  verify  his  thought  by  counting.  Under 
these  circumstances  a  mistake  is  not  to  be  tolerated.  Miss  Smith  emphasizes  the  idea 
strongly  beforehand,  (not,  of  course,  after  the  mistake  has  been  made.  I  hope  you  will 
emphasize  it  in  the  same  way  with  your  pupils.)  In  spite  of  the  teacher's  emphasis  upon 
care  and  upon  "counting  till  you  know,"  Kate,  an  irresponsible  little  creature,  sitting  in  the 
front  row,  blunders.  (Such  things  happen  right  along  as  you  will  find).  Miss  Smith  steps 
over  to  Kate's  desk  and  says  quietly,  "Oh,  I  am  so  sorry.  You  will  have  to  stop."  (Rule 
of  the  game.)  "You  may  watch  Anna's  work."  Anria  is  a  slow,  painstaking  worker.  As 
Kate  sees  her  uncovering  the  numbers  without  a  mistake  she  says  to  herself,  "I  can  do 
that."     The  next  time  the  game  is  played  she  does  do  it. 

Queries.  Why  shoidd  a  teacher  place  her  unsuccessful  pupils  in  the  front 
row  or  as  far  forward  as  possible?  Usually  they  prefer  to  sit  as  far  back  as 
possible.     Why? 

Another  exercise,  "building  ten,"  is  given  with  the  same  purpose  of  helping  the 
pupils  to  a  knowledg:e  of  the  number  symbols  and  their  relative  positions.  The  individual 
charts  are  put  out  of  sight.  The  pupils  arrange  the  little  number  squares  on  their  desks  in 
the  order  of  the  numbers  1 — 10.  If  they  are  uncertain  they  must  refer  to  the  chart.  They 
must  not  make  mistakes  when  it  is  so  easy  to  prevent  them  by  simply  looking  at  the  chart. 
When  they  are  able  to  build  the  first  ten  without  help  from  the  chart,  or  from  any  person, 
they  tc.ke  up  the  next  decade  in  the  same  way.  Miss  Smith  has  wrapped  the  number 
squares  11 — 20  in  papers  of  a  different  color  from  those  which  contained  the  first  ten.  She 
does  this  in  order  that  s\e  may  easily  distinguish  them.  She  keeps  each  set  of  papers  in 
a  separate  marked  envelope  and  the  envelopes  in  a  box  in  the  closet.  Before  giving  out 
the  papers  she  makes  herself  sure  that  the  set  is  perfect.  When  gathering  them  up  she 
appoints  some  one  as  a  committee  to  see  that  none  are  lost.  This  committeeship  is  eagerly 
sought. 

The  preparation  and  care  of  material  and  apparatus  for  number  play  and 
number  work  demand  much  attention  to  detail.  But  it  is  vastly  easier  to  attend 
to  material  and  apparatus  than  it  is  to  teach  number  successfully  without  such 
concrete  aids  to  thinking  in  the  early  stages  of  the  work. 

At  the  time  of  this  writing  there  are  in  Prof,  Baker's  office  a  thousand  each 
of  tags,  test  cards  and  square  feet  of  Manila  paper,  all  cut  to  size  and  waiting 
for  you  to  write  clear,  bold  figures  upon  them,  with  your  cotton-covered  pencil- 
points  or  some  equally  efifective  substitute.  There  are  also  three  new  tagboards 
besides  my  old  one.  These  will  be  passed  around  as  needed.  In  Miss  McCabe's 
office  there  are  three  photographs  on  the  wall  showing  children  working  with 
apparatus.  One  picture  shows  two  little  boys  down  on  the  floor,  "building  tens" 
with  figures  cut  from  calendars.  A  large  piece  of  cambric  has  been  laid  down 
for  them  to  work  upon.  (You  can  use  newspapers  if  you  prefer.)  In  another 
picture  two  others  are  building  the  table  of  fives  on  the  tagboard.  In  the  third 
a  little  fellow  is  building  the  table  of  nines  on  the  buttonboard  while  his  com- 
panion waits  for  the  game  which  will  follow  as  soon  as  it  is  completed.  In  Miss 
Hall's  office  there  is  a  picture  showing  also  children  working  with  test  cards. 
strips,  nailboards,  peg  boxes,  etc.  I  will  show  you  how  to  prepare  and 
use  any  of  these  that  may  seem  desirable  to  work  with  in  your  group.  Such  oc- 
cupations are  more  useful  than  class  drill  in  helping  the  child  to  form  true  con- 
cepts of  the  number  series,  and  as  they  are  really  a  form  of  quiet,  social  play, 
the  children  enjoy  them  greatly,  and  can  engage  in  them  without  fatigue  for  a 


10  ARITHMETIC 

much  longer  time  than  in  ordinary  class  work  or  in  games  where  the  attention  of 
the  whole  class  is  demanded.  Class  exercises,  whether  work  or  play,  should  be 
brief.  Children  have  not  the  power  of  sustained  attention,  and  it  is  worse  than 
useless  to  try  to  hold  their  attention  after  their  stock  of  brain  power  is  exhausted, 
But  in  the  occupations  something  is  to  be  accomplished ;  as,  for  instance,  the  mak- 
ing of  a  "pretty  chart."  The  creative  joy  is  aroused,  concentration  follows,  and 
generally  the  children  are  not  willing  to  stop  until  the  thing  is  finished. 
For  that  reason  you  will  find  it  better  to  put  the  occupations  in  the  latter 
part  of  the  period  after  the  class  exercises.  In  the  case  of  quick  pupils  for  whom 
a  particular  class  exercise  is  not  necessary,  it  is  well  to  leave  them  out  of  it  and 
give  them  an  occupation,  while  you  work  with  the  slow  ones,  mind  to  mind.  Your 
best  and  brightest  pupils  are  likely  to  become  impishly  troublesome  if  they  are 
held  down  to  the  pace  of  the  slow.  (Really  one  can't  help  thinking  that  they 
have  justification.)  They  have  a  right  to  advance  at  their  own  rate  and  it  is 
an  easy  matter  for  a  teacher,  whose  day's  lesson  is  well  planned,  to  give  to  the 
quick  pupils  an  occupation  such  as  written  work  or  chart  work  or  apparatus  work 
in  which  they  are  profitably  employed,  not  merely  occupied,  but  really  learning. 
Such  children  dislike  what  they  call  "baby  work."  "Give  us  something  harder," 
they  often  say.  "I've  got  something  nice  and  hard  to  do  today,"  said  a  little  fellow 
proudly  showing  his  occupation.  Arnold  Bennett,  an  English  writer,  was 
astonished  to  see  children  in  New  York  "grabbing  knowledge  from  their  teach- 
ers," as  he  expressed  it.  If  you  are  successful  in  your  teaching  you  will  see  your 
pupils  "grabbing  knowledge"  from  you  and  from  their  mates.  The  little  ones 
often  have  sweeter  ways  of  getting  knowledge.  When  a  little  child  comes  to  you 
saying,  perhaps  with  a  gentle  tug  at  your  skirt  or  a  soft  touch  on  your  hand, 
"Please  come  show  me  about  this,"  your  first  disengaged  moment  is  the  psycholog- 
ical moment  for  giving  him  the  knowledge  he  wants.  Perhaps  as  you  watch  the  lit- 
tle worker  you  will  feel  an  impulse  to  put  your  arms  around  him  and  give  him  a 
little  hug.  But  if  you  are  wise  you  will  nobly  resist  the  impulse  and  will  keep 
to  the  easy,  pleasant  manner  suitable  to  the  schoolroom.  However,  if  you  feel 
that  impulse,  it  is  probable  that  the  teaching  instinct,  so  nearly  akin  to  the  par- 
ental instincts,  is  welling  up  in  your  mind  and  heart.  This  instinct  for  loving, 
guiding  and  instructing  the  young,  if  reinforced  by  reason  and  good  judgment,  by 
the  study  of  your  pupils  and  also  of  educational  principles  will  bring  you  success 
and  happiness  in  your  work.  Then  you  will  not  become  a  tense,  nerve-racked 
schoolma'am  or  domineering,  dreaded  schoolmaster.  Instead  you  will  be  a  lov- 
ing leader  of  children,  guiding  them  into  the  realms  of  the  world's  knowledge. 

I  have  emphasized  the  fact  that  slow  pupils  should  not  be  prodded  by  blame. 
Quick  pupils  should  not  be  stimulated  and  make  heady  by  praise.  A  child  is  no 
more  deserving  of  praise  for  being  more  capable  that  his  mates  than  for  being 
taller  or  handsomer  or  having  a  better  father  and  mother  than  his  mates.  These 
are  all  matters  of  private  congratulation  but  not  of  public  praise.  The  fact  is 
that  under  those  plans  of  teaching  by  which  children  were  required  to  commit  to 
memory  a  certain  number  of  facts  in  a  certain  time,  whether  they  understood 
them  or  not,  praise  and  blame  were  stimuli  used  to  force  them  into  that  very  dis- 
agreeable form  of  activity,  to  make  them  "get  over  the  ground."  Such  stimuli 
have  no  place  in  our  scheme  of  things,  where  every  child  is  to  be  given  (or  al- 
lowed to  help  himself  to)  all  that  he  can  carry  at  the  time  without  having  any 
dead  weight  of  misconceived  facts  piled  upon  him.  The  quick,  strong,  success- 
ful pupil  should  have  as  a  reward  for  good  work  not  praise,  but  quiet  congratu- 
lation and  the  opportunity  of  acquiring  more  knowledge.  Perhaps  the  teacher 
will  say  to  him  something  like,  "You  got  it,  didn't  you  ?  Tomorrow  you  can  try 
this  piece  of  work.  It  is  still  harder."  Slow,  weak,  unsuccessful  pupils  should 
have  not  blame,  but  sympathy,  expressed  more  in  manner  than  in  words,  recog- 
nition of  their  small  successes,  and  also  sufficient  help.  They  need  frequent  op- 
portunities for  steadying  their  minds  against  the  stronger,  clearer  mind  of  the 
teacher.    They  should  have  direct,  personal  instruction,  lasting  only  a  few  mm- 


OPEN  LETTER  H 

utes,  given  when  the  need  appears.  The  plan  of  giving  your  successful  workers 
special  work  while  you  help  the  unsuccessful,  suggested  on  a  previous  page,  cares 
for  both  kinds  of  workers.  Another  way  of  caring  for  both  kinds  is  to  set  them 
to  working  together.  For  instance,  if  two  pupils,  one  strong  in  his  work  and 
the  other  weak,  build  the  table  of  tens  on  the  tagboard,  working  together,  each  is 
helped  by  the  other.  Especially  is  the  weak  pupil  helped  by  the  strong.  Of 
course  you  understand  that  the  building  of  the  ten  table  is  not  an  end  in  itself, 
that  it  is  merely  a  means  of  putting  the  children  in  contact  with  certain  numbers 
in  a  way  that  brings  into  their  minds  many  vivid  perceptions  of  the  relations  of 
those  numbers.  Care  must  be  taken  to  see  that  the  strong  pupils  are  fair  in  al- 
lowing the  others  to  have  their  full  share  in  the  occupation.  In  this  way  of  work- 
ing there  are  many  beautiful  chances  of  bringing  out  kindly  feelings  towards  the 
child  who  has  missed  work  by  absence  or  is  in  delicate  health  or  is  in  special  need 
of  kindness  for  any  reason.  Of  course  children  are  not  angels  and  sometimes 
complications  arise  from  the  clashings  of  the  wills  of  the  little  people.  In  times 
of  peaceful  activities  the  ideals  of  harmonious  social  work  and  play  are  held  up, 
but  when  conflicts  arise,  the  teacher  promptly  takes  practical  measures  for  secur- 
ing harmony  by  separating  or  perhaps  segregating,  for  a  time,  those  who  "don't 
play  the  game"  with  due  consideration  for  others. 

Educators  generally  agree  that  formal  arithmetic  is  not  suitable  for  children 
of  the  jfirst  and  second  grades.  For  this  reason  it  has  been  taken  out  of  those 
grades  in  many  of  the  best  schools.  But  number  games  and  occupations,  skill- 
fully guided,  are  not  only  pleasing  to  little  children,  but  rightly  used,  they  lead  to 
a  realization  of  numbers  that  cannot  possibly  be  gained  by  the  old-fashioned 
routine  drill  and  enforced  tasks.  Each  game  and  occupation  has  a  specific  pur- 
pose, is  intended  to  lead  the  pupils  to  grasp  some  particular  idea.  The  skillful 
teacher  knows  definitely  what  result  she  wishes  to  obtain  in  giving  it,  just  as 
the  skillful  physician,  administering  a  drug,  knows  the  result  he  desires  to  ob- 
tain by  its  use.  All  the  games  and  occupations  are  carried  on  in 
the  most  leisurely  way,  not  only  to  prevent  errors  in  the  work  but  to  pre- 
vent the  over  stimulation  of  the  children's  minds  and  nervous  systems.  Awful 
things  used  to  be  done  to  the  nervous  systems  of  children  in  schoolrooms  be- 
fore practical  school  work  was  influenced  by  the  kindergarten  movement  and  the 
child  study  movement. 

Queries:  Who  originated  the  kindergarten?  Who  is  the  leader  of  the  child 
study  moz'ement  in  America? 

Every  day  as  the  children  come  back  to  their  happy  social  play  with  the 
numbers  they  gain  a  clearer  idea  of  them  in  their  unchanging  sequence.  Gradu- 
ally the  mental  picture  of  the  series  is  formed  and  becomes  a  part  of  their  mental 
furnishings.  They  can  call  it  up  with  more  or  less  distinctness  just  as  they  call 
up  mental  pictures  of  their  homes  or  school.  A  child  of  six  or  seven  who  has 
lived  for  sometime  in  a  large,  amply  furnished  home  knows  the  forms,  the  loca- 
tions and  the  names  of  hundreds  of  objects  in  the  house.  When  sent  to  get  a 
piece  of  music  lying  on  the  piano  he  does  not  make  the  mistake  of  going  into 
the  kitchen  after  it,  because  his  mental  picture  of  the  interior  of  the  house  locates 
the  piano  in  a  certain  part  of  the  parlor.  No  one  has  drilled  him  upon  the  names, 
places  and  forms  of  the  articles  in  the  house.  He  has  learned  the  forms  and 
places,  without  conscious  effort,  by  contact  with  the  objects  often  repeated.  Ke 
has  learned  their  names  by  auditory  contact,  by  hearing  them  often  applied  to 
the  objects.  And  so  in  our  early  games  and  occupations  we  put  the  child  in  con- 
tact with  numbers,  in  an  agreeable  way,  in  order  that  he  may  learn  their  names, 
their  places  in  the  series,  and  the  forms  of  their  symbols.  While  gaining  this 
knowledge  he  is  unconsciously  learning  many  number  facts,  just  as  by  his  ordin- 
ary play  he  learns  without  effort  many  important  facts  about  this  great  complex 
world  in  which  he  is  an  active  and  inquisitive  newcomer.  Conscious  work  and 
drill  come  later.  Just  now,  you  know,  I  am  writing  to  you  about  beginners  mak- 
ing their  visualizations  of  the  number  series  by  means  of  plays  and  occupations. 


12  ARITHMETIC 

Learning  To  Let  us  go  back  to  the  first  lesson.  In  connection  with  count- 

Apply  Number  ing  by  sound  and  with  chart  work  the  young  teacher  is  aiso 
leading  her  pupils  to  realize  number  by  counting  objects. 
The  children  have  been  in  contact  with  objects  all  their  short  lives,  but  the  idea  of 
number  in  connection  with  them,  except  "two"  or  perhaps  "three,"  does  not  come 
clearly  to  a  child  during  the  first  few  years.  Then  the  number  sense  awakens  and 
gives  him  the  new  enjoyment  of  putting  "the  invisible  band  of  number"  around 
things. 

Miss  Smith's  pupils  count  shoepegs,  button-molds,  pencils,  coins,  inch-squares  and 
triangles  made  by  cutting  inch-squares  diagonally.  They  count  circles  and  half-circles. 
These  geometric  forms  are  arranged  in  patterns.  Sometimes  the  teacher  suggests  the  pat- 
terns, sometimes  the  children  are  allowed  to  use  their  own  taste.  The  counting  idea  is  al- 
ways made  prominent.  Slips  of  paper  upon  which  large  dots  are  arranged  in  forms  like 
those  on  dominoes  are  given  to  them  and  they  report  the  number  of  dots.  (See  page  12  of 
state  textbook  for  illustration.) 

Groups  of  lines,  horizontal,  vertical  or  slanting  are  drawn  on  the  board  by  the 
teacher  or  by  a  pupil  and  the  children  tell  how  many  lines  in  each  group.  They  count  the 
desks  or  tables  in  their  schoolroom.  They  find  the  number  of  pupils  in  the  class.  Some 
day  the  teacher  says,  "Tomorrow  I  want  you  to  tell  me  just  how  many  steps  you  walked 
upon  when  you  came  into  the  school  building." 

At  another  time  they  report  the  number  of  windows  in  a  given  side  of  a  particular 
building  on  the  campus.  Of  course,  not  all  of  the  children  remember  to  count  them,  but 
with  the  consent  of  the  grade  supervisor  the  class  makes  a  pleasant  little  excursion  out- 
side and  every  child  verifies  the  count  of  those  who  did  remember. 

It  is  not  suggested  to  the  pupils  to  count  their  fingers,  and  if  they  begin  to  do  so. 
Miss  Smith  quietly  diverts  their  attention.  She  does  not  wish  them  to  form  the  habit  of 
finger  counting. 

Not  only  things  seen  are  counted  but  things  heard  and  felt.  The  teacher  taps  on 
the  desk  with  a  pencil  and  asks  pupils  to  tell  how  many  taps  were  given.  Turning  to  a 
pupil,  she  says,  "Frank,  you  tap.  Not  more  than  five  times."  He  gives  four  taps.  "Now 
two  more,"  says  the  teacher.    The  class  report. 

At  another  time  she  says,  "I  am  going  to  clap  my  hands.  You  tell  me  how  many  times 
I  clap  them.  Everybody  look  away  from  me."  After  the  pupils  have  responded,  she  calls 
on  members  of  the  class  to  clap  their  hands  and  allows  the  others  to  report  the  number  of 
times. 

She  sets  them  to  marching  around  the  room,  saying,  "Now  you  march  until  I  say  'Halt,' 
and  then  you  must  be  able  to  tell  how  many  steps  you  have  taken." 

She  says  to  the  class,  "I  want  some  one  to  come  out  here  and 
shut  his  eyes  and  see  if  he  can  tell  how  many  shoepegs  I  put  into 
his  hand."  So  uncertain  are  the  reports  of  the  untrained  tactile  sense  that  this 
amounts  to  very  little  more  than  a  guessing  game.  To  prevent  the  class  from  getting  con- 
fined ideas,  she  is  careful  to  have  the  child  whisper  to  her  the  number  of  pegs  that  he 
thinks  he  has  in  his  hand.  She  writes  the  number  and  when  the  pegs  are  shown  and  the 
real  number  found,  the  teacher  merely  says  "Right,"  or  "Wrong." 

Your  own  invention  will  furnish  you  plays  enough  when  you  get  into  the 
spirit  of  the  thing.  The  only  requirements  for  a  good  little  number  play  are 
that  the  children  like  it  and  that  it  teaches  them  something  in  the  right  way. 
Plays  in  which  the  attention  is  not  focused  upon  number,  as  in  some  boisterous 
number  games,  are  objectionable.  "Choosing  sides"  is  not  to  be  recommended 
because  of  the  rivalries  and  jealousies  it  induces,  and  because  as  a  result  of  these 
rivalries  the  zest  of  the  play  depends  upon  the  making  of  mistakes  by  the  oppos- 
ing side.  The  children  listen  eagerly  for  those  mistakes  and  naturally  all  except 
the  clearest  minded  pupils  get  out  of  the  play  an  uncertain  mixture  of  errors  and 
corrections. 

In  the  exercise  just  described  a  group  of  things  was  to  be  observed  by  the  pupils 
and  they  were  required  to  tell  "How  many?"  The  reverse  activity  of  giving  to  the  pupils 
a  number  and  requiring  them  to  select  the  corresponding  number  of  things  is  used  in  many 
ways.  For  instance  the  teacher  or  a  pupil  names  a  number  and  the  class  arrange  on  their 
desks  the  indicated  number  of  squares  or  circles  or  other  objects. 

The  pupils  "tap  out"  or  "clap  out"  numbers  which  the  teacher  or  a  pupil  has  named. 

Sometimes  they  play  a  "mum  game"  with  numbers.  Instead  of  naming  a  number 
the  teacher  or  a  pupil  writes  it  on  the  board  or  holds  up  a  card  upon  which  it  is  pasted. 
The  pupils  tap  it  out ;  not  a  word  is  spoken.  A  mum  game,  played  not  often  and  for  only 
a  short  time,  makes  for  quiet  and  self  control,  but  it  lacks  vitalit}'.  When  Miss  Smith 
sees  that  the  value  of  each  number  in  the  first  decade  is  well  sensed  by  the  children,  the 
plays  are  dropped  and  the  objects  are  put  away. 


OPEN  LETTER  13 

She  uses  a  counting  game  a  little  more  advanced  called,  "Flash."  It  is  played  with 
squares  (or  circles)  of  which  some  are  of  a  dark  color  and  the  others  light.  This  game 
is  useful  only  for  small  groups.  Miss  Smith  calls  up  three  or  four  children  who  stand 
around  her  as  she  sits.  The  remainder  of  the  class  are  at  the  board  writing  numbers. 
Their  turn  comes  later.  She  has  a  boxlid  into  which  she  puts,  perhaps,  three  dark  squares 
and  two  light  ones.  The  children,  with  the  exception  of  John,  whose  back  is  turned  be- 
cause he  is  to  be  the  first  to  play  flash,  watch  her  and  she  consults  them  as  to  how  many 
squares  shall  go  into  the  boxlid.  When  all  is  ready  she  says,  "Flash,  John."  He  turns, 
looks  quickly  and  reports,   "Three   dark  and  two  light.   Five." 

It  will  be  seen  that  here  is  a  little  anticipatory  work  for  the  learning  of 
combinations,  the  securing  of  a  passing  perception  of  the  fact  that  three  and  two 
are  five.  Drill  upon  such  facts  and  enforced  tasks  upon  them  at  this  stage  of 
the  learning  process  would  be  very  harmful.  The  children  need  to  stay  in  the 
concrete  some  time  longer.  Probably  some  child  will  say  with  the  air  of  a  dis- 
coverer, "Yes,  three  and  two  are  always  five."  The  wise  teacher  will  smile  en- 
couragingly at  this  bit  of  generalization,  the  forerunner  of  many  others,  but  she 
will  not  yield  to  the  temptation  to  drill  the  class  upon  it.  There  must  be  many 
such  spontaneous  observations  on  the  part  of  the  children,  much  storing  of  their 
subconsciousness  with  number  facts,  many  resulting  happy  reports  of  their  little 
insights  into  number  before  they  are  ready  for  formal  drill.  Then  they  will  wel- 
come it  as  a  help  in  fixing  in  their  memories  facts  of  which  they  feel  themselves 
to  be  the  discoverers,  but  which,  very  much  to  their  regret,  they  often  lose.  The 
enforcement  of  tasks  will  not  be  necessary.  Voluntary  effort  will  take  the  place 
of  it  in  the  case  of  children  capable  of  learning. 

Does  all  this  seem  to  you  impracticable,  rather  soft-hearted  idealism  ?  It  is 
not.  On  the  contrary  it  is  a  bit  of  the  most  practical,  hard-headed  sort  of  think- 
ing. It  is  not  sentiment  nor  visionary  idealism  that  prevents  Luther  Burbank 
from  tearing  open  rosebuds  in  order  to  get  fine,  full-blown  roses.  His  practical 
knowledge  of  the  laws  that  govern  those  living  organisms  called  plants  leads  him 
instead  to  wait  for  their  development,  in  the  meanwhile  protecting  them  from 
force  and  keeping  them  in  the  most  favorable  conditions  of  soil  and  sun.  And 
it  is  practical  knowledge  of  the  laws  governing  the  minds  of  children  that  leads  a 
teacher  to  wait  for  the  natural,  unforced  development  of  the  mathematical  sense 
of  her  pupils,  keeping  them  in  the  meanwhile  in  the  most  educative  environment 
possible  and  in  tlie  sunshine  of  her  loving  expectancy.  This  is  not  a  new  theory. 
It  is  centuries  old.  In  the  past  its  application  has  been  hindered  by  many  con- 
ditions now  disappearing  under  the  new,  strong  demand  for  efficiency  in  school 
work.  In  differing  forms  but  with  the  same  spirit  the  theory  is  applied  also  in 
other  lines  of  study  here  in  the  Training  School.  You  would  find  it  the  country 
over,  wherever  the  schools  are  controlled  by  advanced  educational  thinking.  Many 
communities  are  "not  yet  ready"  for  that  kind  of  control  of  their  schools,  but  it 
is  only  a  question  of  time  when  they  will  be. 

Running  thru  all  the  work  here  described,  recurring  again  and  agam,  ate 
three  ideas :  Visualization,  Vohmtary  Effort,  and  the  Storing  of  the  Subconscious 
Mind. 

The  visualization  of  the  number  series  secured  by  chart  work  and  apparatus 
work  is  useful  merely  as  a  stepping-stone  to  an  accurate  and  ready  knowledge  of 
number  facts.  This  is  the  ideal  towards  which  we  are  leading  our  pupils, — a 
knowledge  of  number  so  clear  and  strong  that  the  mind  seems  to  respond  auto- 
matically to  any  demands  for  number  facts  such  as  are  used  in  ordinary  calcula- 
tion. For  instance  when  the  question  comes,  "How  much  is  7  times  8  ?"  the  well- 
trained  mind  responds  instantly,  automatically  as  it  were,  with  the  correct  answer. 
It  has  become  a  matter  of  the  reflexes,  like  walking. 

In  the  learning  of  number  facts  there  are  two  periods :  first,  the  period  of 
nnsconcious  learning  by  means  of  play  and  work  with  concrete  material :  second, 
the  period  of  conscious,  voluntary  work  and  play  directed  to  the  memorizing  of 
the  required  facts.  There  are  forty-five  combinations  to  be  learned,  such  as  "4 
and  3  are  ?,"  with  the  corresponding  separations,  as  "4  and  ?  are  7."    In  other 


14  ARITHMETIC 

words,  addition  and  subtraction.  There  are  eleven  multiplication  tables  each  with 
its  eleven  facts,  and  there  are  the  correlative  facts  of  the  division  tables.  It  is 
no  small  task  for  children  with  their  immature  minds  to  perceive  and  fix  in  mem- 
ory these  three  hundred  and  more  number  facts.  It  has  been  called  an  "insuff- 
erably tedious  task."  Fear,  hatred,  and  dread  of  the  work  have  been  supposed  to 
be  the  necessary  accompaniments  of  the  drudgery  it  involved.  But  now  just 
as  you  fortunate  young  people  are  coming  into  the  profession,  plans  for  adapting 
the  work  to  the  capacity  of  the  learner  and  for  utilizing  the  play  spirit  are  be- 
coming general.  As  a  result  the  fear,  the  hatred,  and  the  dread  are  disappearing 
and  interest  and  pleasure  appear.  Some  children  become  too  much  interested  in 
number  work,  just  as  some  people  become  too  much  interested  in  solitaire,  which 
is  after  all  merely  a  grown-up  number  game.  Such  children  should  not  be 
praised  and  pushed  forward.  After  they  have  done  a  fair  amount  of  daily  work 
in  number  their  attention  should  be  diverted  to  other  interests. 

In  the  beginning  of  the  first  period,  the  purpose  of  the  teacher  is  to  arouse 
in  her  pupils  a  sense  of  the  relations  of  numbers  by  means  of  play  and  work  with 
material,  with  charts,  and  with  different  kinds  of  apparatus.  Most  of  the  apparatus 
used  in  our  classes  indicates  number  relations  in  the  same  way  as  the  chart,  but  in 
a  more  objective  form.  During  this  time  children  often  show  that  ideas  of  the  rela- 
tions of  numbers  are  forming  in  their  minds,  by  their  reports  of  their  little  discov- 
eries made  while  using  a  chart  or  some  piece  of  apparatus  or  while  "thinking 
about  it  at  home."  A  child  not  yet  six  years  old  who  had  played  with  a  chart  a  few 
times  called  his  aunt's  attention  to  the  second  horizontal  line  of  the  chart  and  re- 
marked, "All  the  numbers  that  have  2  at  the  end  live  on  the  same  street."  Little 
ones  sometimes  show  reflective  thought  by  such  questions  as  "Where  do  the  num- 
bers go  after  they  get  to  icxd?"  or  "What  is  the  very  biggest  number  in  the  whole 
world  ?"  By  means  of  the  play  and  the  work  and  the  quiet,  happy  thinking  dur- 
ing this  period  the  children  are  led  to  perceive  the  facts  of  number  as  a  series 
of  related  facts.  This  effects  a  great  saving  of  time  and  effort  in  the  later  learn- 
ing process,  as  compared  with  the  plan  of  learning  them  as  independent,  arbi- 
trary statements.  If  in  studying  geography  a  child  should  be  set  to  learning  a 
great  many  separate,  unrelated  facts,  such  as,  "Chicago  is  west  of  New  York,'* 
"Denver  is  east  of  San  Francisco,"  etc.,  without  any  map  to  show  directions  and 
relative  distances,  if  he  simply  memorized  them  as  independent,  arbitrary  state- 
ments, he  might  be  drilled  upon  them  for  a  very  long  time  without  getting  a  clear 
idea  of  the  situation  of  the  places  mentioned  in  these  assertions.  The  facts  would 
be  likely  to  slip  out  of  his  memory  in  the  way  expressed  by  one  of  our  student 
teachers  last  year  when  she  said.  "What  I  teach  these  children  in  the  day  they 
forget  in  the  night."  Now  the  magnitude  of  a  number  is  known  by  its  place  in 
the  number  series.  For  an  illustration  let  us  think  of  the  numbers  ig  and  64. 
From  the  fact  that  ig  comes  earlier  in  the  series  than  64  we  derive  the  idea  that 
it  is  less  than  64.  A  child  playing  with  charts,  number  squares,  etc.,  soon  gets 
similar  ideas,  because  on  the  chart  the  relations  of  numbers  are  shown  by  the 
directions  and  distances  of  the  printed  numbers  from  one  another.  The  chart 
gives  the  same  kind  of  help  in  the  learning  of  number  facts  as  that  which  is  given 
by  a  map  in  the  learning  of  geography.  For  instance,  a  child  in  the  playing  stage 
of  number  learning  sees  that  6,  coming  farther  on  in  the  series  than  4,  is  a  bigger 
number,  means  more  things  than  4.  As  he  becomes  more  definite  in  his  thinking 
he  sees  that  6  is  just  two  steps  beyond  4  or  that  "4  and  two  more  are  6,"  and 
that  "4  and  three  more  are  7."  vSoon  he  sees  that  26  is  just  two  steps  beyond  24 
or  that  "24  and  two  more  are  26,"  and  so  on.  Then  he  is  thinking  number  in- 
telligently. He  is  getting  ideas  of  related  facts  to  be  used  with  clear  percep- 
tion and  strong  memorization  in  his  later  work.  Hence  the  importance  of  this 
first  period  of  play  and  occupation.  It  is  also  important  that  this  period  should 
Come  in  the  early  grades  while  the  child  is  forming  his  concepts  of  number,  at  the 
time  when  his  interest  in  number  and  his  desire  for  childish  play  are  naturally 
strong:. 


OPEN  LETTER  15 

In  the  latter  part  of  this  period  the  pupils  begin  to  acquire  the  power  of 
visualizing  the  number  table  on  the  chart,  of  intentionally  "thinking  how  it  looks." 
At  this  time  there  is  much  quiet,  ruminative  thinking  on  the  part  of  the  successful 
pupils.  They  have  the  visualized  series  at  their  command  and  they  make  many 
observations  upon  it  which  they  report  in  class.  After  the  first  decade  is  mastered 
some  one  is  almost  sure  to  say,  indicating  the  lo,  20,  30,  etc.,  of  the  chart,  "See, 
the  I,  2,  3,  4,  run  right  along  on  the  bottom  line,  too."  As  a  little  fellow  in  one 
of  last  year's  classes  remarked,  while  working  on  the  decades,  "We  are  using 
the  same  old  figures  right  over  again."  When  the  eleven  chart,  given  below,  is 
first  presented,  usually  some  pupil  exclaims,  "The  elevens  all  run  down  hill." 


1 

11 

21 

31 

41 

51 

61 

71 

81 

91 

2 

12 

22 

32 

42 

52 

62 

72 

82 

92 

3 

13 

23 

33 

43 

53 

63 

73 

83 

93 

4 

14 

24 

34 

44 

54 

64 

74 

84 

94 

5 

15 

25 

35 

45 

55 

65 

75 

85 

95 

6 

16 

26 

36 

46 

56 

66 

76 

86 

96 

7 

17 

27 

37 

47 

57 

67 

77 

87 

97 

8 

18 

28 

38 

48 

58 

68 

78 

88 

98 

9 

19 

29 

39 

49 

59 

69 

79 

89 

99 

0 

20 

30 

40 

50 

60 

70 

80 

90 

100 

When  the  nine  chart  appears  they  are  ready  to  see  that  the  multiples  of  nine 
"run  up  hill."  The  spontaneous,  undemanded  expressions  of  the  pupils  in  regard 
to  their  work  are  pretty  good  indications  of  the  vitality  and  success  of  your  own 
eflForts  in  teaching. 

As  you  have  been  in  Prof.  Baker's  classes  and  have  studied  his  manual,  you 
know  the  value  he  places  upon  correct  mental  picturing  at  this  stage  of  the  work. 
The  usefulness  of  these  mental  pictures  is  also  clearly  recognized  in  a  recent  book, 
The  Teaching  of  Arithmetic,  by  Dr.  A.  W.  Stamper  of  the  Chico  Normal  School. 

In  this  visualizing  work  it  is  the  aim  of  the  teacher  to  make  her  pupils  ab- 
solutely independent  of  all  concrete  aids  to  thinking.  Gradually  she  leads  them 
from  the  use  of  the  chart  to  the  use  of  mental  imagery.  "I  wish  I  had  a  chart 
like  this,  at  hom.e.  Then  I  could  reckon  anything  I  wanted  to,"  said  a  boy  in 
the  second  grade,  beginning  number.  "You  don't  need  it,"  replied  the  teacher. 
"You  will  soon  have  one  in  your  head."  Her  prophecy  came  true.  In  a  few 
months  the  boy  was  able,  while  standing  with  his  back  to  the  chart,  to  answer 
such  questions  as  "How  many  are  96  and  3?"  "50  and  20?"  He  answered  these 
as  correctly,  tho  not  as  quickly,  as  he  would  have  answered  the  question  "How 
many  windows  are  there  in  your  mother's  kitchen?"  and  by  the  same  process, 
that  of  forming  a  mental  picture. 

I  hope  it  is  clear  to  you  that  this  visualizing  stage  in  which  the  learner  brings 
up  a  mental  picture  of  the  number  series  and  upon  it  counts  out  facts  just  as  he 
formerly  counted  them  out  on  the  chart,  is  only  a  passing  phase  of  the  learning 
process.     It  is  intermediate  between  the  counting  on  the  chart  and  the  quick, 


16  ARITHMETIC 

sure  knowing  of  the  required  number  facts.  After  that  is  reached  there  is  no 
need  of  visuahzations  of  the  number  series.  If  any  exist  they  He  unused  in  the 
background  of  consciousness,  just  as  the  mental  picture  of  the  typewriter  key- 
board, so  necessary  at  first,  is  unused  by  the  typist  after  she  becomes  expert.  But 
while  the  child  is  in  the  visualizing  stage  we  must  see  to  it  that  his  mental  pic- 
tures are  correct  and  vivid.  There  are  several  little  exercises  designed  to  secure 
good  visualizations,  which  I  will  show  you  in  your  classrooms.  None  of  them, 
however,  are  as  effective  as  the  simple  device  of  keeping  a  plainly  written  number 
chart  always  on  the  board  in  convenient  range  of  the  pupils'  sight.  The  silent, 
unremitting  instruction,  sent  by  the  chart  into  the  minds  of  the  pupils  when- 
ever they  chance  to  turn  their  eyes  towards  it,  greatly  shortens  the  time  and 
effort  necessary  for  the  conscious  learning  of  number.  As  Prof.  Baker  expresses 
it,  the  children  are  "exposed"  to  mathematical  truth.  Much  is  absorbed  by  their 
impressionable  minds,  and  it  is  stored  in  the  subconsciouness. 

The  second  period  of  number  learning,  that  in  which  the  child  is  con- 
sciously acquiring  the  knowledge  of  number  facts  that  is  to  serve  him  the  rest 
of  his  life,  lasts  two  or  three  years.  At  this  time  the  purpose  of  the  teacher  is 
to  help  her  pupils  to  become  so  prompt  and  accurate  in  their  responses  to  num- 
ber questions  that  their  mental  action  seems  to  be  automatic.  The  time  in  which 
this  power  of  reflex  response  to  demand  for  number  facts  can  be  acquired  dif- 
fers greatly  in  individuals.  It  cannot  be  shortened  by  outside  force  as  that 
only  produces  bewilderment.  Hence  the  need  of  waiting  for  the  individual 
ability  to  develop.  Some  children,  not  well  equipped  by  nature,  can  not  be- 
come quick  and  accurate  in  number  work  except  at  an  expense  of  time  and  effort 
greatly  disproportionate  to  its  value.  For  them  the  "minimum  essentials"  mast- 
ered and  managed  in  their  own  slow  way  are  all  that  are  feasible.  The  power 
to  reckon  quickly  and  accurately  is  soon  lost  by  disuse  just  as  facility  in  piano- 
playing  or  in  typewriting  is  soon  lost  when  practice  stops.  Of  course  it  may 
be  regained  by  renewed  practice.  This  power  of  ready  response  to  number 
questions  can  never  be  gained  by  a  learner  whose  perceptions  of  number  facts 
are  vague  or  incorrect.  Hence  the  importance  of  vivid  and  accurate  presenta- 
tions of  those  facts. 

There  is  a  theory  well  known  to  educators,  the  theory  of  "brain  patns," 
given  by  Prof.  William  James  in  his  great  work.  Principles  of  Psychology.  He 
presented  it  as  a  hypothesis  used  to  coordinate  many  known  facts  about  the 
way  in  which  people  learn.  Psychologists  agree  that  all  of  our  thinking  is  ac- 
companied by  motion  of  the  molecules  of  certain  cells  in  our  brains,  and  that  this 
motion  passes  along  from  one  cell  to  another.  Hence  the  name  "path."  In  the 
light  of  this  theory  we  shall  see  that  it  is  our  bounden  duty  to  insist,  from  the 
beginning  and  all  thru  the  work,  upon  absolute  correctness,  secured  by  slow, 
clear  perceptions  and  by  careful  expression.  According  to  this  theory  the  first 
time  a  child  thinks,  "5  and  3  are  8,"  he  starts  a  brain  path.  Every  repetition  of 
that  fact  deepens  and  smooths  the  path,  as  it  Avere,  and  makes  the  recalling  of 
the  fact  more  easy  and  swift,  until  after  many  repetitions  the  action  of  the  bratn 
centers  involved  becomes  reflex.  But  if  he  makes  the  statement  that  5  and  3 
are  something  else  than  8,  or  hears  it  made  by  his  mates,  a  new,  diverging  brain 
path  is  made,  along  which  impulses  are  likely  to  travel  whenever  the  question 
is  asked,  "How  many  are  5  and  3  ?"  Uncertainty  begins,  uncertainty  that  some- 
times lasts  until  the  pupil  has  reached  high  school  work,  as  many,  many  teach- 
ers besides  myself  can  sadly  testify.  To  prevent  this  uncertainty,  or  any  kind  of 
mistaken  thinking  about  number  facts,  it  will  be  necessary  for  you  to  insist  that 
every  child,  when  in  doubt,  shall  take  the  truth  at  once  from  the  visible  number 
series  of  the  chart,  without  giving  time  for  any  wrong  impulses  to  play  thru 
his  brain.  We  say  to  our  pupils.  "If  you  think  mistakes  when  you  are  little  it 
will  hurt  your  brains  and  make  you  stupid  in  arithmetic  when  you  get  older, 
and  if  you  say  mistakes  out  loud  you  will  hurt  the  other  children's  brains."  This  is 
exactly  true,  and  we  find  it  very  effective  in  arousing  in  the  pupils'  minds  a  strong 


OPEN  LETTER  17 

aversion  to  mistakes.  In  this  way  the  "storing  of  the  subconsciousness"  with 
errors  can  be  prevented  to  a  certain  extent.  Of  course  it  will  be  impossible  for 
you  or  anyone  else  to  prevent  it  entirely. 

When  the  work  of  memorization  goes  on  successfully  the  interest  heightens. 
The  joy  of  acquisition  is  aroused  in  the  children.  They  rejoice  in  the  possession 
of  their  bits  of  num.ber  knowledge  as  the  miser  counting  his  coins,  rejoices  in  his 
possessions.  They  express  their  feelings  in  such  remarks  as,  "I  know  the  even 
numbers  and  I  know  the  table  of  tens  and  I  know  a  lot  of  the  fives."  Sometimes 
they  ask  for  drill  for  some  particular  purpose.  For  instance,  a  child  who  wants 
to  get  more  fives  or  to  keep  what  she  has  got  will  perhaps  ask  the  teacher  for  a 
class  drill  upon  the  table  of  fives.  It  is  a  useful  plan  to  have  the  pupils  make  in 
class  little  books  in  which  they  write  the  number  facts  that  they  are  sure 
they  know.  These  little  books,  made  by  folding  a  few  sheets  of  paper, 
and  having  a  title  like,  ''My  First  Number  Book,"  should  of  course  be  as  neatly 
made  as  possible.  When  filled  they  may  have  a  cover  decorated  according  to 
the  fancy  of  the  owner  or  the  taste  of  the  teacher.  They  should  be  exactly  true 
in  their  contents.  For  this  reason  the  pupils  should  put  into  the  books  only  a 
very  few  facts  at  a  time  and  those  should  be  facts  upon  which  they  have  "stood 
test"  several  times.  Of  course  the  teacher  will  carefully  examine  the  books  to 
see  that  no  errors  creep  in.  If  any  are  found  the  teacher  should  erase  them 
without  mentioning  them.  In  the  next  class  period  she  will  question  the  pupil 
about  the  fact,  and  when  she  is  satisfied  that  he  has  it  clearly  in  m.ind  she  will 
allow  him.  to  put  the  statement  of  it  in  the  vacant  space. 

My  dear  students,  in  this  letter  many  topics  are  touched  upon,  ranging  from 
playthings  to  psychological  theories,  and  I  know  that  many  of  the  ideas  will  not 
be  clear  to  you  until  they  have  been  elaborated  in  your  professor's  classroom 
and  in  our  weekly  conferences  or  until  they  have  been  worked  out  in  your  own 
classrooms.  If  you  will  read  again,  and  as  a  continuous  story,  the  descriptive 
work  printed  in  small  type,  you  will  see  that  it  merely  covers,  in  sketchy  outline, 
the  learning  of  the  three  first  essentials  mentioned  on  page  3,  the  sound  series, 
the  sight  series,  and  their  use  in  applying  number  to  objects.  Next  comes  the 
understanding  of  the  decimal  notation  with  its  recurrences  and  repetitions,  and 
along  with  it  the  writing  of  numbers  beyond  100.  It  is  not  worth  while  to 
present  any  more  descriptions  in  this  letter  to  you,  because  a  few  minutes  of 
demonstration  work,  such  as  I  hope  to  give  in  your  classrooms  from  time  to 
time,  will  help  you  more  than  dozens  of  pages  of  description  of  it,  besides  being 
much  easier  for  you,  and  for  me.  The  book  which  I  am  writing  about  the  happy 
learning  of  mathematics,  a  copy  of  which  I  hope  to  place  in  our  library  in  the 
course  of  a  year  or  two,  contains  a  great  deal  of  description  of  schoolroom 
activities.  It  is  not  the  purpose  of  this  letter  to  give  you  a  set  of  pedagogical 
devices.  Its  aim  is  to  lead  you  to  think  intelligently  about  the  principles,  pur- 
poses, and  reasons  for  procedure,  that  underlie  the  work.  Plans,  methods, 
devices,  apparatus,  all  avail  little  or  nothing  unless  the  true  spirit  and  under- 
standing are  in  the  teacher.  When  these  come  to  you  and  with  them  the  skill 
to  adapt  the  work  to  your  own  classes  you  will  find  many  plans  useful  to  you  in 
books,  and  especially  in  school  journals. 

The  learning  of  the  facts  and  processes  of  addition,  subtraction,  multipli- 
cation, and  division  is  the  principal  work  of  the  third  and  fourth  grades.  Dur- 
ing this  time  the  pupils  are  also  forming  clear,  elementary  ideas  of  multiples, 
factors,  fractions,  ratios,  measurements,  and  geom.etnc  forms  as  presented  in  the 
drillbook  used  in  the  third  grade.  They  are  also  learning  something  about  the 
applications  of  numbers  in  daily  life.  As  soon  as  they  have  a  few  number  tacts 
they  begin  to  use  them  in  number  stories.  (See  page  29  of  state  textbook  for 
illustration.)  They  play  store,  count  money,  and  make  change.  They  report 
actual  purchases  that  they  have  made.  Suppose  Hugh  has  bought  a  pencil,  cost- 
ing 5  cents.  The  class  may  tell  the  different  coins  used  in  making  change  if  a 
quarter  was  offered  in  payment  or  a  half  dollar,  or  a  dollar.    Care  nuist  be  taken 


18  ARITHMETIC 

that  the  children  do  not  make  disclosures  of  family  affairs  in  their  eagerness  to 
report  actual  buyings  in  which  they  are  interested.  Other  applications  of  number 
may  be  drawn  from  their  work  in  manual  training,  in  construction,  and  in  other 
departments,  but  the  main  stress  should  be  laid  upon  the  direct  memorizing  of  facts. 
This  is  to  be  enlivened  by  many  number  games  which  will  be  shown  to  you. 

No  formal  analysis,  no  problems  requiring  the  consideration  of  "steps,"  should 
be  given  at  this  time.  This  is  the  time  when  the  perceptive  and  retentive  powers 
are  to  be  used,  not  the  powers  of  formal,  abstract  reasoning.  In  the  training  of 
children  for  circus  performers  the  exercises  are  carefully  adapted  to  their  powers. 
Apart  from  considerations  of  humanity  the  trainers  are  unwilling  that  a  child 
should  be  overtrained  or  made  fearful,  as  that  would  probably  prevent  him  from 
becoming  a  fine  adult  performer.  Long  ago  Pres.  Eliot  pointed  out  the  fact  that 
if  children  are  to  become  clear-thinking  mathematicians  in  their  later  years  of 
learning,  they  must  not  be  given,  in  their  early  years,  exercises  that  involve  be- 
wilderment and  struggle.  Many  of  the  failures  in  the  higher  grades  are  due  to  this 
cause.  Many  pupils  enter  those  grades  who,  instead  of  having  a  clear  knowledge 
of  number  which  they  can  use  in  connection  with  the  reasoning  processes,  have  an 
abounding  hatred  of  it  as  the  chief  result  of  their  early  unsuccessful  struggles  with 
the  subject. 

In  the  pre-memorizing  period  of  working  with  the  comibinations  the  child  has 
them  all  before  him  on  the  chart,  and  he  uses  them  in  various  ways ;  as  in  the  exer- 
cise of  "telling  combinations",  in  which  each  child  has  his  turn  in  selecting  com- 
binations from  the  chart  or,  if  he  is  very  sure  of  himself,  from  his  mind.  But  when 
the  work  of  memorizing  begins,  the  combinations  are  not  to  be  presented  in  a  hap- 
hazard way.  Instead,  a  few  are  selected  and  they  are  thoroughly  learned  and 
presented  as  a  stunt  by  each  child.  Games,  drills,  tests,  boardwork,  bookwork,  all 
bear  upon  the  set  of  combinations  which  is  being  considered.  In  this  way  the 
desire  of  the  pupils  to  acquire  it  is  aroused.  And  their  desire  is  the  great  im- 
pelling force  in  all  this  zvork. 

The  first  set  of  combinations  consists  of  those  of  the  even  numbers.  It  is 
preceded  by  counting  by  twos,  and  by  games  with  even  numbers.  Let  me  caution 
you,  if  you  want  to  avoid  mix-ups,  don't  use  the  word  "odd"  in  connection  with 
the  odd  numbers  at  this  time.  The  numbers  are  even  or  not  even.  INIonths  later 
when  the  distinction  is  clear  in  the  minds  of  the  children,  it  will  be  safe  to  use  the 
word  "odd".  When  pupils  are  able  to  add  2  to  any  even  number  that  they  can  see 
on  the  chart  or  can  think  of,  they  practice  adding  10  to  even  numbers  as  well  as  2. 
It  is  easy  to  add  10  on  the  chart,  as  it  requires  only  a  move  to  the  right  on  the 
horizontal  line,  a  fact  which  the  pupils  soon  discover.  4  is  taken  next.  They  then 
have  the  three  numbers,  2,  10,  and  4,  which  they  use  as  addends  with  other  even 
num.bers.  They  naturally  want  more.  6  is  given  to  them  and  they  are  set  to  ring- 
ing the  changes  of  these  four  numbers  upon  the  even  numbers.  When  they  are 
expert  in  this,  they  take  8.  This  completes  the  first  set  of  combinations.  Then 
come  the  corresponding  separations.  The  use  of  flash  cards,  of  test  cards,  and 
much  written  work  is  needed.  Column  addition  of  even  numbers  and  the  addition 
of  two  numbers  in  the  thousands,  all  of  whose  digits  are  even  numbers,  are  used 
at  this  time.  Keen-minded  children  soon  discover  a  law  which  one  expressed  as 
follows :  "If  you  add  an  even  number  to  another  even  number  you  will  get  an  even 
number,  and  if  you  keep  on,  you  will  always  get  even  numbers.  The  other  num- 
bers are  not  so."  A  little  girl  remarked,  "The  other  numbers  are  just  put  in  between 
the  even  numbers  to  hold  them  together."  Upon  this  a  boy  said  that  he  thought 
"they  were  put  there  to  keep  the  even  numbers  apart."  The  teacher  could  not 
decide,  but  she  was  greatly  pleased  with  these  comments  because  they  showed  that 
the  attention  of  the  children  was  focused  upon  the  even  numbers,  which  was  just 
where  she  wanted  it  to  be  at  that  particular  time.  When  the  class  has  stopped 
blundering  or  doubting  on  this  set  of  combinations  and  separations,  it  is  ready  for 
the  second  set.  Probably  some  individuals,  who  got  ready  before  the  class  in  gen- 
eral, are  already  preparing  stunts  in  advance. 


OPEN  LETTER  19 

The  second  set  consists  of  the  combinations  that  make  the  even  numbers  thru 
i8,  group  combinations,  as  4  equals  2  and  2,  and  also  3  and  i,  etc.  I  hope  that 
those  of  you  who  have  this  work  this  year  will  remind  me  to  give  you  the  game 
"How  many?" 

The  third  set  includes  all  the  other  combinations ;  that  is,  the  groups  that  make 
the  odd  numbers  thru  17. 

It  will  take  many  months  for  the  pupils  to  master  the  combinations  and  sep- 
arations. If  the  teacher  hurries  and  crowds  the  work,  the  pupils  will  become  con- 
fused in  their  thinking  and  it  will  take  many  more  months.  It  is  a  common  mis- 
take of  inexperienced  teachers  to  suppose  that  when  a  pupil  has  recited  a  fact  he 
will  continue  to  know  it.  The  first  learning  is  always  temporary  memorization, 
scarcely  more  than  perception.  There  must  be  many  repetitions  of  it  at  different 
times  before  the  brain  path  is  worn  deep  enough  to  be  permanent. 

I  do  not  believe  that  you  are  able  to  tell  what  combination  of  foods  composed 
your  dinner  a  week  ago  last  Thursday.  You  had  a  vivid  knowledge  of  it  at  the 
time.  All  your  senses  reported  it  to  you.  (Observe  that  in  learning  number  only 
visual,  auditory,  tactile,  and  motor  impulses  are  involved.)  But  I  venture  to  say 
that  your  knowledge,  vivid  as  it  was  at  the  time,  has  proved  to  be  only  a  "tem- 
porary memorization,"  If  you  were  to  have  the  same  combination  of  foods  at 
many  meals,  your  memorization  of  them  would  be  more  lasting,  its  permanence 
depending  upon  the  number  of  the  repetitions. 

Query.  V/liat  does  this  illustration  suggest  to  you? 

During  the  months  in  which  the  facts  of  addition  and  subtraction  are  the  cen- 
tral thought  of  the  work,  the  facts  and  processes  of  multiplication  and  division  are 
also  being  comprehended  and  to  some  extent  memorized.  The  multiplication  tables 
are  taken  in  the  following  order :  tens,  twos,  fives,  elevens,  nines,  threes,  eights, 
fours,  sevens,  sixes,  twelves.  The  reasons  for  this  order  will  appear  in  the  work. 
They  are  too  lengthy  to  be  given  here. 

It  is  a  fact  well  known  to  teachers  of  mathematics  that  pupils  can  carry  on 
two  or  more  lines  of  activity  at  the  same  time  without  becoming  confused,  pro- 
vided that  one  does  not  depend  upon  another.  For  instance,  if  we  were  to  intro- 
duce the  second  set  of  combinations  before  the  first  was  mastered,  confusion  would 
result ;  but  when  we  allow  the  pupils  to  work  on  a  multiplication  table,  it  does  not 
interfere  with  the  learning  of  the  combinations  and  it  is  a  pleasing  bit  of  advance. 
A  day's  lesson,  which  lasts  forty  minutes  in  these  grades,  may  include,  besides 
combinations,  work  and  play  with  a  multiplication  table  and  with  geometric  forms 
in  which  doubling,  halving,  and  otherwise  varying  the  forms  is  a  feature.  On 
another  day  simple  ratios  will  be  presented.  The  pupils  can  see  from  the  chart 
that  10  is  one  half  of  20,  just  as  one  column  is  one  half  of  two  equal  columns. 
When  the  ideas  of  one  third,  one  fourth,  etc.  are  grasped  in  the  work  with  geomet- 
ric forms,  the  fact  that  10  is  one  third  of  30,  one  fourth  of  40,  and  so  on  is  equally 
clear.  The  day's  lesson  should  usually  include  a  little  new  work  carefully  adapted 
to  the  children's  comprehension.  As  an  opposite  kind  of  activity,  some  snappy 
work  for  speed,  judiciously  used,  is  helpful  and  enjoyable.  It  must  not  be  carried 
to  the  point  of  over  excitement.  The  last  twenty  minutes  of  the  period  is  de- 
voted to  written  work  from  the  book.  Then  is  the  teacher's  opportunity  to  call  to 
her  some  pupil  who  needs  a  few  minutes  of  help.  Perhaps  there  is  a  boy  who  is 
"a  problem,"  indifferent,  careless,  out  of  harmony.  As  he  stands  by  the  teacher's 
knee,  as  he  would  by  that  of  his  mother  or  father,  and  receives  the  friendly  help, 
he  may  become  interested,  perhaps  softened  by  the  mind-to-mind  contact  over  the 
work.    It  is  worth  trying. 

About  plans.  You  know,  of  course,  that  a  teacher  who  goes  before  her  class 
without  a  definite  purpose  to  teach  some  particular  thing  or  without  tentative  plans 
for  teaching  it,  is  a  failure  from,  the  start,  just  as  your  dressmaker  or  tailor  would 
be  if  attempting  to  make  a  suit  for  you  without  having  definite  plans  for  it.  Your 
daily  plans  should  be  made  on  the  day  before  they  are  used.  For  instance,  in  your 
preparation  period  on  Monday,  while  Monday's  lesson  and  its  immediate  results 


20  ARITHMETIC 

are  still  fresh  in  your  mind,  think  over  the  plans  for  the  next  day's  lesson,  select 
the  exercises  that  you  propose  to  use  and  make  a  list  of  them.  Have  this  list  for 
the  day  on  your  table,  where  it  may  be  consulted  by  the  grade  supervisor  and  my- 
self when  we  enter  your  room.  We  trust  that  it  will  be  read  occasionally  by  the 
principal  and  by  the  head  of  the  mathematics  department,  who  are  also  interested 
in  your  work. 

Instead  of  a  written  plan  for  the  coming  week  please  hand  to  me  at  the 
weekly  meetings  a  written  report  of  what  you  have  done  in  the  past  week  and 
the  results  of  it  as  far  as  you  are  able  to  determine. 

As  soon  as  you  have  decided  upon  the  seating  of  your  group,  please  make  a 
diagram  of  it,  giving  the  full  names  of  the  pupils  in  the  order  in  which  they  sit 
and  your  own  name  and  grade.  Hand  this  to  me  at  your  first  opportunity. 

In  your  few  weeks  of  practice  study  in  the  Normal  you  are  workmg  under 
favorable  conditions  created  especially  to  enable  you  to  learn  the  ways  in  which 
children's  minds  react  under  teaching.  Many  of  the  responsibilities  which  will  be 
yours  in  your  future  work  in  the  schoolroom,  here  devolve  upon  your  super- 
visors. In  order  that  we  may  be  helpful  to  you  and  may  protect  your  pupils 
from  any  harm  that  might  come  to  them  from  the  unguided  efforts  of  inex- 
perienced teachers,  it  is  necessary  for  us  to  know  your  work  as  closely  as  possible. 
As  your  success  means  the  success  of  the  children  and  also  our  success,  nr.my 
earnest  and  kind  wishes  are  centered  upon  you. 

With  pleasant  anticipations  of  our  work  together,  I  am 

Your  sincere  friend  and  co-worker, 

Ade;lia  R.  Hornbrook. 
San  Jose,  Cal.,  Sept.,  1913. 


ERRATA  FOR  "OPEN  LETTER" 

Page     4,  47th  line,  5th  word,  "deficiencies." 
Page  10,  39th  line,  8th  word,    "made." 

Page  19,  1st  line,  "rest  of  the"  should   be   between   the   6th   and   the   7th 
word. 


III. 

THE  FIRST  FIVE  YEARS 
OF  ARITHMETIC 

Second  Letter  From  a  Supervisor 

Copyright,  1914,  By  Adelia  R.  Hornbrook. 


My  dear  Student  Teachers  : 

It  is  important  that  each  of  yon  shall  understand  the  general  course  of  in- 
struction of  which  your  term's  work  is  a  component  part.  This  chapter  contains 
a  brief  outline  of  the  number  play  and  work  of  the  first  two  years  and  of  the 
more  formal  work  of  the  three  succeeding  years  at  the  end  of  which  the  ele- 
mentary state  textbook  is  completed.  It  also  contains  many  suggestions  for  work 
which  heretofore  have  been  given  orally. 

First  Twenty  minutes  a  day  are  given  to  number  play  consisting  of  occu- 

Year  pations  with  simple  apparatus,  plays,  and  games.  The  general  purposes 
of  the  first  year  activities  are  to  lead  the  child  to  know  the  sequences  and  values 
of  the  first  hundred  numbers,  to  give  him  motive  and  opportunity  to  perceive  and 
use  many  independent  number  facts,  to  help  him  to  memorize  the  combinations 
whose  sum  is  lo  or  less,  and  to  enable  him  to  read  and  write  numbers  as  far  as 
thousands. 

As  the  opportunity  for  voluntary  eft'ort  is  always  open  to  the  children  some 
of  them  learn  and  present  as  "stunts'"  much  more  than  others.  At  the  end  of 
each  term  the  classes  are  sectioned  so  that  the  quick  pupils  are  in  one  group  and 
the  slow  are  in  another,  no  idea  of  inferiority  or  superiority  being  attached  to 
either  set  of  pupils.  If  you  are  put  in  charge  of  a  section  of  slow  pupils,  you  will 
have  the  opportunity  to  use  much  sympathy,  watchfulness,  and  ingenuity.  It 
is  often  the  case  that  the  "slow"  pupil  is  merely  the  unawakened.  If  the  awak- 
ening of  the  mathematical  sense  comes  while  he  is  under  your  care,  you  will 
have  the  pleasure  of  assisting  in  rapid  and  happy  development. 

In  the  "Open  Letter"  which  forms  the  preceding  chapter  of  this  manual  some 
beginnings  of  the  plays,  occupations  and  games  of  the  first  year  are  described. 
(See  pp.  3,  4,  5,  6,  7,  8,  9,  12,  13.)  These  descriptions  are  given  as  a  basis  for 
your  work.  Follow  them  closely  at  first  until  your  teaching  sense  begins  to  de- 
velop.   Then  vary  the  plans  to  suit  the  occasions  that  arise. 

Each  exercise  has  its  specific  purpose.  For  instance,  such  plays  as  "hunt- 
ing", "hiding",  and  "letting  out"  numbers,  described  on  pages  7  and  9,  are  de- 
signed to  teach  the  child  how  to  read  numbers.  They  also  help  him  to  realize 
the  value  of  numbers  by  fixing  in  his  mind  their  places  in  the  series,  as  shown 
in  the  chart. 

For  the  beginning  work  desk  charts  are  to  be  prepared,  one  for  each  child. 
(See  description  p.  8.)  In  the  IB  grade,  only  the  first  decade  should  be  writ- 
ten in  the  desk  charts.  The  second  is  added  after  the  first  is  learned,  and  so  on. 
Keep  a  large  chart  of  fives  always  upon  the  board.  The  other  apparatus  must 
be  prepared  when  needed,  except  tagboard,  buttonboard,  nailboard,  decade  sticks, 
abacus,  etc.,  which  are  furnished  by  the  school. 

In  all  grades  each  teacher  before  beginning  work  should  place  her  plan  for 
the  day's  lesson  upon  the  supervisor's  desk  and  should  see  that  all  the  material 
required  in  it  is  at  hand.  It  is  as  absurd  for  a  teacher  to  go  before  her  class  with 
only  her  voice  for  a  tool  as  it  would  be  for  a  carpenter  to  attempt  to  work  with 
only  his  fingers,  having  neglected  to  bring  his  tools.  Among  the  tools  of  the 
teachers  of  the  first  and  second  grades  are  charts,  number  squares,  crayons,  eras- 

21 


22  ARITHMETIC 

ers,  pencils,  paper,  large  cards  with  figures  for  tracing,  small  cards  with  figures 
for  games,  cards  with  domino  spots,  flash  cards,  buttonstrings,  dissected  charts, 
rulers,  rolls  of  ribbon  paper,  objects  for  counting  and  combining,  as  squares,  cir- 
cles, semicircles,  triangles,  star  points,  leaves  of  trees  (especially  the  peppertree), 
toothpicks,  buttonmolds,  shoenails,  etc.  Charts,  crayon  and  erasers  are  akvays 
needed.  From  the  others  the  teacher  should  choose  a  sufficient  number  of  those 
needed  to  carry  out  the  plan  of  the  day's  work. 

Fresh  air  is  so  important  in  the  schoolroom  and  our  soft  climate  allows 
us  to  obtain  it  so  easily  that  your  first  care  upon  entering  your  classroom,  what- 
ever may  be  your  grade,  should  be  to  see  that  your  windows  are  rightly  adjusted. 

Definite  plans  and  an  abundance  of  material,  altho  they  are  essential, 
are  not  enough  to  insure  success.  I  have  seen  in  the  classroom  most  dreary,  per- 
functory, futile  exercises  which  were  sincerely  intended  for  number  play  by  the 
mistaken  young  teacher  who  carried  them  on.  She  had  her  plan  and  her  material 
but  at  least  two  great  essentials  for  success  were  lacking  in  her  mind.  The  first 
was  the  play  spirit.  Without  the  spirit  of  play  all  this  number  play  is  worse 
than  useless.  "'How  can  I  get  the  play  spirit?"  do  you  ask?  It  can  be  acquired 
in  various  ways.  Watch  children  at  play  and  try  to  interpret  their  actions,  play 
with  them,  recall  your  own  childish  plays  and  feelings,  talk  with  those  who  play 
with  children  successfully,  as  kindergartners  and  playground  supervisors.  Read 
books  and  articles  upon  the  subject.  For  your  immediate  purposes  when  you  are 
appointed  to  a  number  play  section,  if  you  take  into  your  classroom  the  plays 
suggested  in  this  manual  and  shown  in  the  supervisor's  meetings  and  enter  into 
them  wholeheartedly,  watching  the  children,  thinking  about  them,  trying  to  in- 
terpret their  feelings,  you  will  find  the  spirit  of  play  coming  to  you.  It  is  natural 
for  children  to  love  to  play.  And  it  is  just  as  natural  for  women  and  For  many 
men  to  love  to  play  with  children. 

There  must  also  be  in  the  mind  of  the  teacher  clear  ideas  of  numbers  in 
their  relations  to  one  another  and  a  strong  desire  to  lead  her  pupils  to  realize 
certain  bits  of  number  knowledge  by  means  of  the  happy,  purposeful  play.  This 
desire  will  lead  her  to  watch  the  actions  of  her  pupils'  minds  and  to  devise  means 
for  reaching  them.  She  will  strive  to  adjust  the  olay  to  the  needs  of  every  little 
member  of  her  flock. 

Every  term  I  watch  with  delight  the  development  of  teaching  power  m 
students  who,  after  a  period  of  apparent  daze  and  weak,  uncertain  effort,  catch 
the  spirit  of  the  play,  and  as  they  grasp  its  underlying  purposes  become  so  filled 
with  desire,  determination  and  expectancy  for  their  pupils'  happy  learning  that 
not  only  is  the  success  of  the  pupils  assured  but  their  own.  As  I  write,  a  mental 
picture  of  a  scene  in  the  classroom  of  one  of  these  successful  young  teachers 
comes  before  me.  She  was  a  quiet,  soft-voiced  woman.  There  was  no  affected 
sweetness  nor  forced  jollity  in  her  manner.  She  was  deeply  interested  in  her 
work  and  acted  naturally.  In  her  directions  to  her  class  she  was  clear,  confident 
and  exact.  She  did  not  harangue  her  children,  nor  shout  orders  at  them,  nor  ex- 
hort them  to  "be  good"  or  "pay  attention."  As  I  entered  the  room  one  day  she 
was  sitting  at  one  side  of  the  large  table  with  the  children.  They  were  playing 
with  the  abacus.  (Ordinarily  an  abacus  has  twelve  rows  of  beads,  but  ours  have 
been  made  to  correspond  with  the  chart  by  clipping  the  beads  from  two  of  the 
wires.)  She  was  carrying  the  children's  attention  in  an  easy  way.  She  was  not 
discouraged  by  the  lapses  of  the  little  brains.  She  merely  led  the  wandering  at- 
tention back  to  the  desired  point  by  directly  and  pleasantly  including  the  inat- 
tentive little  one  in  the  play.  In  the  course  of  it  a  boy,  whom  I  will  call  John, 
announced,  "When  you  have  ten  beads  on  the  abacus  and  take  ten  more  you  have 
twenty,  just  like  the  chart.  When  you  have  ten  numbers  on  the  chart  and  then 
ten  more  you  have  twenty."  He  made  his  meaning  clear  by  showing  the  first 
two  decades  on  the  chart  and  pointing  out  the  20.  "Did  all  of  you  hear  what 
John  said?"  asked  the  teacher.  "Tell  it  again,  John."  John  told  it  again  more 
definitely  and  with  mor^  delight  than  before. 


FIRST  FIVE  YEARS  23 

The  teacher  had  planned  other  things  for  that  lesson,  but  like  the  wise 
little  woman  that  she  was,  she  seized  the  opportunity  thus  given.  She  led  the 
class  into  a  discussion  of  '"what  John  found  out."  They  showed  on  their  desk 
charts  the  fact  that  2  tens  make  20.  John  proudly  handled  the  abacus,  that 
privilege  being  the  natural  reward  of  the  discoverer.  It  was  developed  also  that 
3  tens  are  30,  and  that  4  tens  are  40,  ("Don't  you  see  the  4  in  the  40?"  was 
asked,)  and  so  on.  Not  all  the  children  grasped  the  idea  that  day  but  it  got  into 
their  mental  atmosphere.  Sooner  or  later  as  the  subject  came  up  they  all  realized 
the  facts  and  hence  remembered  them  in  a  way  impossible  to  them  if  they  had 
been  set  to  memorizing  the  table  as  a  mere  patter  of  words. 

When  a  child  sees  for  himself  that  the  3  in  30  means  3  tens,  that  6  tens  are 
6-ty,  7  tens  are  7-ty,  etc.,  then  (and  not  until  then)  is  he  ready  to  drop  the  use 
of  the  concrete  and  to  take  up  with  ease  the  conscious  learning  of  the  multiplica- 
tion table  of  tens.  "Standing  the  test"  upon  this  table  and  later  upon  other 
tables  are  achievements  in  which  children  take  much  rightful  pride. 

Note  carefully  that  what  is  meant  by  the  "table  of  tens",  as  presented  to  be- 
ginners, is  the  series  of  statements,  "i  ten  is  10,"  "2  tens  are  20,"  eic. 
To  the  children  each  ten  is  an  entity,  a  ten  of  beads  on  the  abacus  or  a  ten  of 
circles  on  their  table  or  of  numbers  on  the  chart.  A  statement  about  5  tens  has 
for  them  as  concrete  a  basis  as  a  statement  about  5  sticks.  The  reversed  ten 
table,  "10  times  i  are  10,"  "10  times  2  are  20,"  etc.,  does  not  deal  with  tens. 
It  deals  with  ones,  twos,  threes,  etc.  This  reversed  ten  table  is  of  course  neces- 
sary to  be  known,  and  we  lead  up  to  it  by  a  gradual  process  of  reversing  the 
known  facts  of  one  table  so  as  to  correlate  it  with  the  others,  or  as  we  call  it  in 
the  classes,  by  "finding  old  friends"  in  a  new  table.  In  all  grades  where  the  chil- 
dren are  learning  the  tables  of  multiplication  and  of  division,  teachers  should 
be  sure  that  they  are  using  the  table  for  which  the  children's  concrete  work  has 
prepared  them ;  otherwise  the  repetition  of  the  statements  becomes  mere  word 
patter. 

I  know  that  you  are  handicapped  in  the  number  play  by  the  fear  that 
the  discipline  of  your  room  will  not  be  satisfactory.  But  your  department  super- 
visors are  eager  to  have  the  children  led  into  knowledge  of  number  by  organized, 
well-directed  play.  They  comment  happily  and  enthusiastically  upon  the  suc- 
cess of  those  student  teachers  that  accomplish  it.  It  is  necessary  to  distinguish 
between  the  informal  play  by  which  children  learn  number,  and  the  formal  dis- 
cipline by  which  they  enter  and  leave  the  room,  or  execute  any  concerted  move- 
ments. This  formal  discipline  is  not  only  a  necessity  in  the  managing  of  class- 
es of  children  but,  rightly  used,  it  is  a  great  benefit  to  the  children  individually 
in  many  ways.  One  has  only  to  look  at  a  West  Point  graduate  to  see  some  of 
the  physical  benefits  of  discipline.  The  personal  and  social  ideals  inculcated  iu 
quiet,  prompt,  orderly,  concerted  action  in  the  schoolroom  are  of  great  value. 
You  can  get  this  discipline  by  first  finding  out  just  what  it  ought  to  be  and  then 
insisting  upon  it  from  the  first  in  a  kind,  absolutely  decided  way,  taking  if  for 
granted  that  you  have  the  sweet  obedience  and  cooperation  of  your  pupils  in 
"doing  things  right."  You  must  be  able  to  pass  at  will  from  formal  discipline 
to  informal  play  and  back  again.  Without  this  foundation  of  discipline,  the  de- 
tails of  which  you  must  get  as  soon  as  possible  from  the  department  supervisors, 
the  play  becomes  wild  and  purposeless,  very  tiring  to  the  children  (and  teacher), 
and  doing  more  harm  than  good  to  their  number  sense.  Observe  and  use  from 
the  first  the  signal  by  which  a  child  signifies  his  readiness  to  speak,  the  "po- 
sition". This  is  for  several  reasons  a  great  improvement  upon  the  old  plan  of 
raising  hands. 

All  the  plays  are  very  simple  and  are  based  upon  well-known  psycholog- 
ical principles  which  it  is  not  necessary  to  discuss  here.  They  make  varied  ap- 
peals to  the  senses  and  to  the  capacities  of  the  child.  He  not  only  hears,  he  sees, 
feels,  handles,  constructs,  chooses,  counts,  measures,  draws,  matches  and  ad- 


24  ARITHMETIC 

justs,  observes,  discovers  and  reports,  gives  "stunts",  keeps  scores  and  reports 
them,  tells  stories  and  comments  upon  those  of  his  mates,  dramatizes,  in  short, 
uses  his  mind,  his  body  and  the  number  series  in  many  interesting  ways.  And 
thru  it  all  he  has  the  joy  of  acquisition,  of  feeling  that  he  is  getting  knowledge 
of  number, — that  wonderful  thing  that  grown-ups  seem  to  know  so  much  about 
and  that  will  help  him  to  be  like  them  when  he  learns  it.  None  of  the  plays  are 
boisterous  altho  there  is  much  movement,  sometimes  rhythmic,  as  in  point- 
ing and  counting,  or  marching  and  singing,  and  sometimes  free,  as  when  the 
children  walk  over  to  the  corners  of  the  table  and  count  out  for  themselves  the 
number  of  circles  or  triangles  or  other  pieces  needed  for  the  patterns  which  they 
are  making.  This  is  much  better  than  having  them  sit  passively  while  you  pass 
out  the  pieces,  provided  that  the  children  are  quiet  and  polite.  It  is  well  to  tell 
them  that  this  nice  way  of  getting  the  pieces  can  only  be  used  in  rooms  where 
the  children  are  polite.  It  is  well  to  mention  the  same  thing  casually  to  some 
grown  person  (as  for  instance  a  supervisor)  in  the  presence  of  the  children. 

Absolute  accuracy  is  insisted  upon  as  an  essential  part  of  the  plays.  (See 
the  story  of  Tom,  p.  5,  also  theory  of  "brain  paths",  p.  16.)  It  is  vastly  better 
for  children  to  be  playing  outside  than  to  be  in  the  classroom  learning  false  state- 
ments about  numbers.  And  if  they  are  allowed  to  make  errors  and  to  hear  them 
they  will  learn  untrue  statements.  Of  course  you  will  not  repeat  an  error  nor 
call  attention  to  it.    Quickly  and  emphatically  substitute  the  fact. 

Among  plays  suitable  for  first  grade  pupils  are  Matching  Numbers  on  the 
chart ;  Having  Numbers  Dance ;  Locating  Numbers ;  Counting  and  Pointing  on 
the  chart  by  tens,  or  by  fives  or  twos ;  Building  Tens  on  the  table  with  number 
squares,  or  blank  squares,  circles,  semicircles  or  triangles ;  Building  Tens  with 
tags  on  the  decade  sticks,  or  on  the  tagboard,  or  with  buttons  on  the  button- 
board,  with  beads  on  the  abacus,  or  with  nails  on  the  nailboard.  After  all  these 
building  occupations  the  game  of  "Telling  Which"  is  useful,  because  it  causes 
definite  perceptions  of  the  places  occupied  by  different  numbers  in  the  series  and 
leads  to  clear  visualizations.  When  the  desk  charts  become  slightly  worn  they 
are  replaced  by  new  ones  and  the  old  charts  are  cut  into  strips  each  showing 
a  decade.  Putting  these  decades  together  in  their  original  form  is  one  way  of 
Building  a  Hundred.  Again  the  old  charts  are  cut  into  strips  each  showing 
five  numbers.  These  strips  are  used  to  build  a  hundred.  Or  the  charts  are  cut 
into  twos  with  which  to  build  twenty  or  thirty  or  fifty,  as  the  children  may  be 
able  in  the  given  time.  Some  multiple  of  10  is  always  chosen  as  the  limit  of  the 
building.  After  a  time  the  system  of  decimal  notation  seems  to  dawn  upon  the 
children.    Then  they  are  ready  for  new  ideas  and  new  plays  to  lead  up  to  them. 

Running  to  a  Number,  begun  in  the  first  grade,  is  used  with  adaptations  in 
the  first  five  grades.  This  begins  with  "creeping."  For  instance,  the  teacher 
directs  the  children  to  put  their  left  forefingers  upon  3  on  their  desk  charts. 
"Count  on  four,"  she  says.  With  the  left  finger  still  on  3  they  count  and  point 
out  with  the  right  the  next  four  numbers  and  report  "7."  They  hold  the  rigTit 
finger  upon  the  7  until  another  addend  is  given.  Then  the  left  finger  is  brought 
forward  in  the  counting  and  pointing  and  is  placed  upon  the  number  reported. 
And  so  on  until  a  desired  number  is  reached.  The  teacher  uses  "Count  on"  and 
"Add"  interchangeably  until  the  former  is  dropped.  Soon  the  children  are  able 
to  count  without  pointing  out  each  number.  Then  the  small  fingers  no  longer 
"creep"  but  "walk"  over  the  chart.  Later  when  the  children  have  learned  the 
distances  and  directions  of  the  respective  numbers  the  fingers  "run",  occasionally 
"jumping",  as  when  ten  or  some  multiple  of  ten  is  added.  The  children  take 
turns  acting  as  leaders  in  giving  addends.  But  first  the  teacher  leads,  usually 
carrying  some  one  idea  thru  the  exercise.  Perhaps  she  begins  with  6,  adds  4, 
then  6,  then  4  and  so  on  until  some  child  sees  and  reports,  "I  don't  have  to  count. 
When  you  add  4  to  a  number  that  has  6  at  the  end  it  always  brings  you  down  to 
the  end  of  the  decade  where  there  is  a  0."  Of  course  he  is  invited  to  show  the 
"easy  way"  to  his  classmates.    Some  of  them  see  it  at  once.    Perhaps  some  slow- 


FIRST  FIVE  YEARS  25 

minded  little  one  will  report  the  same  thing  somie  time  afterward  with  great 
delight,  believing  it  to  be  his  own  discovery.  Such  a  belated  perception  is 
warmly  welcomed  by  the  teacher  when  it  does  come. 

To  give  a  foretaste  of  subtraction  the  teacher  directs  the  children  to  "run 
backward"  on  the  chart.  Then  she  sends  them  forward  again,  giving  such  direc- 
tions as  "Put  finger  on  5.    Tell  how  many  you  must  add  to  reach  8." 

Later  the  pupils  "run  to  numbers"  on  the  wall  chart.  In  the  succeeding 
grades,  after  the  mental  picture  of  the  number  series  is  distinct  (See  p.  15), 
they  run  "in  their  minds".  Still  later  when  they  reach  the  stage  of  automatic 
knowing,  towards  which  all  this  childish  but  clear  and  happy  perception  work 
is  designed  to  lead  (See  p.  13),  the  exercise  has  become  for  them  really  ac- 
curate and  rapid  addition  and  subtraction.  Other  ways  of  anticipating  addition 
and  subtraction  are  by  means  of  various  plays  with  tagboard,  buttonboard,  etc. 

To  make  the  pupils  familiar  with  the  combinations  that  make  10  is  the  pur- 
pose of  several  of  the  early  plays.  The  "parting  game",  plays  with  the  button- 
string,  "fishing  for  tens",  and  the  game  "How  many?"  are  among  those  used 
to  give  practical  ideas  of  group  combinations,  not  only  of  10.  but  later  of  other 
numbers.    The  footrule  is  also  used  in  finding  group  combinations. 

When  your  pupils  play  these  games  quickly  and  surely,  prepare  a  set  of  test- 
ing cards  and  call  for  volunteers  to  "stand  test"  upon  the  combinations  that  make 
10.  Then  the  separations.  Keep  records.  Later  let  them  try  the  combinations 
of  other  numbers.  Remember  that  in  every  test  "the  first  mistake  spoils  the 
stunt". 

In  "Making  Patterns"  definite  numbers  of  geometric  forms,  as  circles,  half- 
circles,  squares,  triangles,  rhombuses,  etc.,  are  combined  into  regular  figures 
either  copied  or  original,  the  number  idea  being  made  prominent. 

The  time  during  which  the  digits  are  presented  (See  pp.  7  and  8),  should 
be  at  least  three  months.  As  the  children  learn  to  make  the  figures  correctly, 
the  making  of  charts  upon  the  board  or  upon  squared  paper  becomes  a  useful 
and  pleasing  occupation.  After  the  pupils  know  the  first  hundred  numbers  and 
you  think  that  they  are  ready  for  larger  ones,  if  you  will  place  a  tag  showing  25 
over  the  two  naughts  on  the  100  tag,  usually  some  one  will  be  able  to  read  the 
125  for  you.  If  there  is  little  or  no  response,  wait  a  few  weeks  before  trying  it 
again.  Do  not  let  any  one  say  "One  hundred  and  twenty-five."  Omit  the  "and." 
The  children  play  with  even  numbers  in  many  ways.  They  march  and  count 
by  twos,  sing  the  even  numbers,  sort  out  even  numbers  written  on  cards,  group 
objects  in  pairs  and  report  the  number  of  pairs,  make  two-charts  with  the  first 
thirty  numbers,  play  the  game,  "Hunting  Even  Numbers,"  in  which  two  chil- 
dren go  to  corners  of  the  room  and  hide  their  eyes  while  the  rest  select  an  even 
number  for  them  to  find  from,  hints  given  them,  or  the  game  in  which  they  "fish 
for  even  numbers."  Running  to  numbers  stepping  only  on  the  evens,  is  a  use- 
ful exercise.  Plays  v/ith  even  numbers  have  many  variations  but  they  all  have 
the  same  purpose — to  focus  the  child's  thought  upon  the  even  numbers  to  the  ex- 
clusion of  the  others  during  this  time  of  preparation  for  the  later  work  of  learn- 
ing the  first  set  of  CQmbinations  and  the  table  of  twos.  Hence  in  playing  with 
even  numbers  the  odd  numbers  are  never  intentionally  mentioned. 

Gradually  the  children  begin  to  remember  some  of  the  facts  which  they  are 
perceiving  and  reporting  in  their  plays.  Then  they  are  ready  for  the  exercise, 
"Giving  Number  Facts".  The  phase  of  development  in  which  children  are  eager 
to  give  out  number  facts  which  they  have  subconsciously  stored,  appears  usually 
in  the  lA  or  2B  grade.  At  first  the  teacher  writes  upon  the  board  the  facts 
given  by  the  children,  thus  showing  them  the  correct  ways  of  writing  their  be- 
loved facts.  Soon,  however,  they  write  them  upon  the  board,  upon  paper  and  in 
little  blank  books  made  by  folding  a  sheet  of  foolscap.  Filling  a  Number  Fact 
Book  is  a  fine  occupation  for  those  who  are  ready  for  the  work.  When  the 
teacher  sees  an  error  on  the  board,  she  erases  it  and  sets  the  pupil  to  verifying 


26  ARITHMETIC 

his  fact  by  "counting  it  out"  on  his  chart.  Papers  containing  errors  are  not  re- 
turned to  the  pupils.  When  the  teacher  finds  a  mistake  in  a  number  fact  book 
she  erases  it,  or  if  it  is  written  in  ink  she  cuts  it  out. 

Second  Number  play  and  number  work  go  on  in  a  leisurely  way  thru  the 

Year  second  grade,  twenty  minutes  a  day  being  spent  upon  it.     The  play 

is  lessened  as  its  purposes  are  attained.  The  work  is  increased,  but  it  is  always 
adjusted  to  the  stage  of  "readiness"  of  the  class.  The  early  occupations  give  way 
to  such  interesting  exercises  as  working  examples,  applying  numbers  to  geomet- 
ric forms,  discovering  the  fractional  parts  of  surfaces,  lines,  solids,  and  num- 
bers. The  general  purpose  is  to  lead  the  pupils  to  memorize  the  first  set  of  com- 
binations and  separations  (those  of  the  even  numbers),  and  the  tables  of  tens, 
fives,  twos  and  elevens,  and  to  learn  to  read  and  write  numbers  to  millions. 

There  are  usually  some  children  in  a  class  whose  mental  age  is  beyond  that 
of  their  mates.  A  child  of  seven  may  have  the  mental  power  of  an  ordinary  child 
of  ten.  Such  children  must  not  be  held  down  to  the  work  of  their  mates  which 
they  already  know  but  must  be  allowed  to  learn  and  present  as  much  as  the  time 
permits.  Their  work  is  not  only  a  joy  to  themselves  but  it  is  an  inspiration  to 
their  mates.  In  forming  the  habit  of  independent,  advancing  work  they  are  lay- 
ing the  foundation  of  later  success.  While  the  teacher  should  not  stimulate  them 
she  should  give  them  from  time  to  time  the  bits  of  information  which  they  may 
need.    This  applies  in  all  grades. 

Giving  number  facts  is  an  important  exercise  in  the  2B  grade.  The  chil- 
dren get  their  facts  from  the  chart  or  from  their  mental  pictures  of  it  and  from 
one  another.  Sometimes  a  child  presents  a  particularly  enormous  fact  about  which 
he  says  "  I  got  it  from  papa."  (Of  course  3'ou  will  look  out  for  the  weak  ones  and 
see  that  every  one  has  some  fact  to  present.)  The  little  people  like  to  deal  with 
large  numbers.  Heiice  it  becom.es  necessary  to  show  them  how  to  read  and  write 
thousands,  and  later  millions.  The  Million  Stick  and  the  metal  numbers  are  used 
for  these  purposes.  Most  pupils  are  eager  to  take  up  "easy  ways"  of  getting  num- 
ber facts.  For  instance,  they  see  that  10  can  be  added  to  any  number  without 
counting,  by  "just  going  straight  across  to  the  next  decade".  The  teacher  shows 
them  such  facts  as  that  in  adding  7  to  40,  instead  of  finding  40  on  the  chart  and 
counting  on  seven  numbers  to  47,  as  they  have  been  accustomed,  they  can  just 
put  a  7  in  the  place  of  the  o  in  the  40.  Quick  pupils  see  at  once  this  short  way 
of  adding  numbers  less  than  ten  to  the  multiples  of  ten,  and  the  subject  is  brought 
up  from  time  to  time  until  all  grasp  the  idea.  Doubling  Numbers  and  its  com.- 
plement,  Halving  Even  Numbers,  furnish  many  facts.  When  a  child  knows 
from  his  own  experience  that  5  and  5  are  10,  it  is  not  difficult  for  him  to  see  that 
5  is  one  half  of  10.  If  you  have  the  children  place  on  their  desks  an  odd  number 
of  circles,  as  five,  and  tell  them  to  find  half  of  the  circles  it  Avill  be  interesting  to 
see  how  many  of  them  grasp  the  idea  of  halving  a  number  that  is  not  even. 
Be  sure  to  put  the  results  of  such  trials  into  your  weekly  report. 

Column  addition,  the  addition  of  numbers  in  the  thousands,  working  with- 
out the  chart,  and  other  "grown-up"  ways  of  dealing  with  numbers  are  very  pleas- 
ing to  the  little  learners  at  this  stage.  Addition  with  "carrying"  is  so  desirable 
an  accom.plishment  that  sometimes  they  get  it  from  their  elders  at  home.  Before 
they  have  learned  carrying,  it  is  necessary  to  use  for  addends  only  those  numbers 
in  which  the  sum  in  each  order  except  the  highest  is  less  than  ten,  as  826  with 
843,  It  is  a  good  plan  to  call  upon  som.e  child  to  give  the  first  addend,  supplying 
the  other  addend  yourself.  But  after  they  have  learned  to  carry,  as  they  can  get 
all  needed  facts  from  their  charts,  it  is  well  to  let  them  furnish  both  addends  ex- 
cept when  you  have  some  special  point  which  you  wish  to  bring  out. 

Efface  your  own  activities  and  promote  those  of  the  children  as  much  as 
possible  but  always  keep  a  watchful  eye  and  a  guiding  hand  upon  them. 

After  the  children  have  given  number  facts  of  their  own  choosing  for  a  time 


FIRST  FIVE  YEARS  27 

they  are  ready  to  begin  memorizing  in  regular  order  the  facts  of  addition  and  sub- 
traction. For  convenience  these  are  divided  into  three  sets.  The  first  set  consists 
of  combinations  in  which  both  addends  are  even  numbers.  In  the  second  set  both 
addends  are  odd  numbers.  In  the  third  set  one  of  the  addends  is  an  even  num- 
ber and  the  other  odd.  For  a  detailed  plan  for  teaching  the  first  set  of  combina- 
tions and  separations  see  page  i8. 

Observe  that  from  this  time  on  there  are  in  each  lesson  two  kinds  of  work, 
perception  work  and  memorization,  and  that  they  are  managed  quite  differently. 
In  the  perception  work  the  children  are  not  required  to  furnish  from  memory  the 
facts  needed  in  the  exercises.  They  use  their  charts  freely  altho  they  are 
proud  when  they  can  say  after  finishing  a  piece  of  work,  ''I  did  it  without  my 
chart."  But  in  this  early  memorizing  work  the  children  are  to  be  led  to  de- 
pend upon  the  memory.  For  instance,  when  a  class  has  learned  to  add  lo,  2,  and 
4  to  each  even  number  on  the  chart,  the  teacher,  before  beginning  that  part  of 
the  day's  lesson  in  which  this  particular  set  of  facts  is  used  in  working  examples 
or  in  tests,  pins  a  large  piece  of  paper  over  the  wall  chart  and  the  children  turn 
their  desk  charts  over  upon  their  desks.  This  is  done  as  a  necessary  preliminary 
to  the  new  and  important  kind  of  work,  the  memorizing.  In  case  of  a  failure 
there  is  quick  reference  to  the  chart,  but  the  ideal  of  successful  work  is  that  of 
sure  (and  later  quick)  memory  work.  The  perception  work  gives  pleasant  and 
thoro  preparation  for  the  conscious,  definite  progress  which  is  the  real  aim  of 
all  these  efforts.    Gradually  as  its  purpose  is  accomplished  it  is  dropped. 

Until  the  first  set  of  additions  and  subtractions  are  so  well  mastered  that 
every  child,  except  those  who  are  to  be  retarded,  has  "stood  test"  upon  them  and 
can  use  them  accurately,  the  children  do  not  begin  to  memorize  those  of  the  sec- 
ond set,  usually  not  before  the  third  grade.  If  a  child  fails  upon  a  test  which  he 
passed  triumphantly  a  few  weeks  before,  do  not  blame  yourself,  nor  your  prede- 
cessors, nor  the  child.  The  constitution  of  the  ordinary  human  mind  is  such  that 
there  must  be  many  temporary  memorizations,  many  forgettings  and  renewings, 
before  our  knowledge  of  mathematical  facts  is  permanent  and  readily  available. 
It  is  the  business  of  the  school  to  furnish  opportunities  for  these  renewings,  with- 
out haste  or  impatience,  simply  dealing  with  the  mathematical  nature  of  the  child 
as  it  actually  is,  instead  of  assuming  that  it  is  what  one  might  wish  it  to  be. 

As  helps  in  the  conscious  memorizing  we  use  the  Identification  Game,  the 
Testing  Game,  played  by  partners,  and  Individual  Tests.  For  these,  exercises, 
cards  must  be  prepared.  Flash  cards,  which  are  very  useful,  can  be  bought.  As 
the  list  of  successful  test-passers  grows,  call  for  volunteers  to  try  the  speed  test 
with  cards  or  to  be  "it"  in  the  game  of  "Catch  m.e  if  you  can." 

Thruout  the  year,  work  and  play  with  the  multiplication  and  division  tables 
is  carried  on  parallel  with  that  of  addition  and  subtraction.  One  line  of  effort  is 
stressed  for  a  week  or  two,  or  until  certain  results  appear,  and  then  the  other. 

Knowledge  of  the  table  of  tens  is  a  great  help  in  learning  the  table  of  fives. 
The  children  sing  the  table  of  fives,  pointing  out  the  multiples,  or  they  sing  and 
march.  They  make  five-pointed  stars  out  of  pieces  of  colored  paper,  each  of  which 
is  a  symmetrical  fifth  of  such  a  star.  Reversing  the  little  trapeziums  that  form  the 
stars,  they  make  ten-pointed  figures.  They  are  led  to  see  and  report  such  facts 
as  that  when  they  have  twenty  star  points  they  can  make  four  of  the  five-pointed 
stars  or  two  of  the  stars  that  have  ten  points.  Building  oblongs  with  inch 
squares,  making  fans  with  toothpicks  on  the  desks,  reckoning  pansy-petals  (or 
those  of  any  other  five-petaled  flower),  are  useful  exercises.  Pupils  use  the  mul- 
tiple squares,  build  the  table  on  a  tagboard  or  show  it  on  a  buttonboard,  or  play 
"Greeting  the  Multiples."  (We  use  the  word  "multiple"  in  its  objective  sense, 
of  course,  and  it  is  no  more  difficult  for  children  than  "automobile"  or  "radi- 
ator." To  them  it  means  "big,  bright  number  on  the  chart.")  They  recite  the 
table  in  order  forward  and  backward,  first  in  unison  and  then  as  individual 
"stunts".    Then  come  the  games  with  flash  cards,  the  identification  game  and  the 


28  ARITHMETIC 

various  kinds  of  tests.  Many  applications  of  the  table  of  fives  are  made  in  num- 
ber stories  about  nickels,  dimes,  etc.  Some  pupils  like  to  play  store,  count  money 
and  make  change  and  we  offer  them  the  chance,  but  that  work  goes  better  in  the 
third  grade.  Making  clock-faces  and  reckoning  minutes  by  fives  are  useful  ex- 
ercises. 

The  same  general  plans  are  used  in  the  teaching  of  all  the  tables,  but  each 
table  has  its  own  special  applications.  The  table  of  twos  gives  many  opportu- 
nities for  reckoning  pairs,  as  gloves,  shoes,  hands,  feet,  eyes.  The  children  pour 
water  from  pint  bottles  into  quart  bottles  and  back  again  and  discuss  the  num- 
ber of  pints  and  quarts  of  imaginary  milk  necessary  to  supply  certain  numbers 
of  imaginary  families.  Before  a  new  table  is  taken  up  the  preceding  ones  are 
reviewed.  The  table  of  elevens  (See  page  15)  is  shown  to  the  pupils  as  a  flight 
of  stairs  down  which  they  first  walk,  then  run.  It  is  very  easily  learned  because 
pupils  soon  see  such  facts  as  that  the  fourth  step  is  made  of  two  fours,  the  sev- 
enth step  of  two  sevens  and  so  on.  Here  we  begin  the  correlation  of  the  tables 
by  telling  the  children  that  there  is  in  this  table  an  "old  friend"  that  we  met  in 
the  five  table,  and  setting  them  to  hunt  for  it.  When  they  have  found  55  perhaps 
some  one  will  report  the  finding  of  an  old  friend  from  the  two  table,  22. 

In  the  last  part  of  the  second  year  pupils  begin  progressive  written  work  with 
individual  advance.  This  is  a  favorite  occupation.  A  set  of  lesson  sheets  carefully 
graded,  containing  work  which  they  have  memorized,  is  prepared.  In  the  work 
upon  these  sheets  all  begin  together  upon  Lesson  i.  If,  upon  examining  the  pa- 
pers, the  teacher  finds  an  error,  she  cuts  it  out.  At  the  succeeding  lessons  the 
work  of  each  child  begins  where  the  perfect  work  of  his  last  writing  ended. 
When  he  has  finished  Lesson  i,  in  perfection  without  using  the  chart,  he  is  al- 
lowed to  take  Lesson  2,  and  so  on.  A  record  of  each  child's  progress  is  kept. 
After  a  time  the  children  become  separated  in  their  individual  work,  some  per- 
haps reaching  Lesson  6  or  7  while  others  are  still  working  upon  Lesson  2.  This 
process  of  natural  selection  aids  in  the  sectioning  of  the  grade  at  the  end  of  the 
term. 

Third  Forty  minutes  a  day.    The  new  work  of  this  grade  is  the  first  mem- 

Year  orizing  of  the  remaining  facts  of  addition  and  subtraction,  also  of  the 
tables  of  nines,  threes,  eights,  fours,  sevens,  and  sixes. 

The  first  work  is  the  thoro  reviewing  and  testing  of  the  first  set  of  combina- 
tions and  separations  and  of  the  tables  of  tens,  fives,  twos  and  elevens.  New  ap- 
plications of  these  tables  are  made,  as  in  dramatizing  and  in  the  Battle  Game  and 
in  the  finding  of  new  fractional  parts  of  the  oblongs  and  other  geometrical  figures. 
Written  multiplication  is  taught  in  connection  with  these  tables.  At  first  exam- 
ples are  given  where  no  carrying  is  involved,  such  as  the  multiplying  of  222  or  of 
521  by  2,  3,  and  4.  Later  the  pupils  take  such  examples  as  the  miultiplying  of 
115  or  251  or  502  by  each  of  the  numbers  from  2  to  9  inclusive.  Notice  that  the 
multiplicands  contain  only  the  figures  i,  2,  o,  and  5,  and  require  knowledge  of 
only  the  tables  that  are  being  reviewed. 

Written  subtraction  is  taught  as  the  finding  of  a  missing  addend,  by  the 
method  so  clearly  explained  on  pages  37-40  of  the  California  state  elementary 
arithmetic. 

In  order  to  prevent  the  repetition  of  mistakes,  see  that  each  child  in  your 
class  has  his  correction-book,  made  by  folding  a  sheet  of  paper,  and  that  when- 
ever he  makes  an  error  in  reciting,  he  immediately  puts  the  correct  statement 
into  his  book.  Occasionally  drill  upon  these  statements.  You  will  find  probably 
that  there  are  certain  facts  upon  which  many  pupils  are  apt  to  fail.  Of  course 
you  will  emphasize  these  facts. 

As  soon  as  the  children  are  able  to  read  problems  intelligently,  a  primary 
number  book  is  used  as  a  basis  for  progressive  written  work.  Before  the  books 
are  given  out,  while  the  lesson  sheets  are  still  in  use,  the  teacher  prepares  her 
class  for  the  new  difficulties  of  the  book  by  teaching  its  first  subjects  in  an  easy 


FIRST  FIVE  YEARS  29 

conversational  way.  In  this  teaching  and  in  the  work  with  the  book  the  pupil  is 
free  to  consult  his  chart  at  any  time  when  his  memory  fails  to  supply  promptly 
the  desired  fact.  By  this  habit  of  carefully  referring  instead  of  guessing,  mis- 
takes are  avoided  and  the  facts  are  fixed  in  the  mind.  While  the  pupils  are  writ- 
ing, the  teacher  ought  not  to  move  around  among  them.  She  should  sit  quietly  in 
front  of  her  class  near  the  board.  If  a  pupil  needs  help  he  should  come  forward 
after  obtaining  permission  and  put  his  problem  on  the  board.  Encourage  your 
pupils  to  work  independently  of  you,  as  much  as  possible. 

The  same  plans  of  "teaching  ahead"  with  the  whole  class,  of  testing,  and  of 
writing  with  individual  advance  are  used  in  the  succeeding  grades.  Without 
this  careful  and  successful  preliminary  teaching  the  work  in  the  book  becomes 
difficult  and  confusing. 

In  teaching  the  second  set  of  combinations  and  subtractions,  use  the  same 
general  plan  as  in  teaching  the  first  set.  Begin  with  3  and  use  it  as  an  addend 
with  numbers  whose  unit  figure  is  i,  then  3,  5,  7,  9.  As  the  additions  are  learned 
practice  the  corresponding  subtractions,  the  finding  of  missing  addends.  Give 
tests  and  applications.  When  the  children  generally  are  sure  and  prompt  in  add- 
ing 3  to  the  odd  numbers  (and  not  before)  use  5  as  an  addend  in  the  same  way, 
then  7,  then  9.  Test  and  apply  at  every  stage,  making  the  pupils  feel  that  ac- 
curacy and  progress  are  the  two  delightful  things  in  their  work  and  that  their 
progress  depends  upon  their  accuracy.  The  first  and  second  sets  together  include 
all  the  group  combinations  that  form  even  numbers.  Practice  these  combina- 
tions thru  18  thoroly  before  taking  up  the  third  set.  If  the  memorizing 
of  the  first  and  second  set  of  additions  has  been  well  done,  that  of  the  third  is 
comparatively  easy.  To  each  of  the  numbers  i,  3,  5,  7  and  9  practice  adding  2  un- 
til the  pupils  neither  blunder  nor  hesitate.  In  the  same  way  and  with  repeated 
tests  and  applications  use  as  addends  4,  6,  and  8.  As  a  review,  reverse  the  pro- 
cess, and  to  each  of  the  even  numbers  2,  4,  6,  8,  add  first  3,  then  5,  7,  9.  These 
form  the  group  combinations  of  the  odd  numbers  thru  17,  and  they  should  be 
faithfully  practiced. 

The  table  of  nines  on  the  chart  presents  a  long  flight  of  steps  to  be  ascended 
and  also  the  short,  easy  flight  made  of  90  and  99.  There  are  four  "old  friends" 
in  this  table  and  the  children  should  be  allowed  to  hunt  for  them.  The  square 
yard  furnishes  a  special  application  for  the  table  of  nines.  Draw  a  square  yard  on 
the  board  and  divide  it  into  square  feet.  Let  the  children  renew  it  as  needed. 
Use  it  for  a,  basis  for  number  stories  and  problems.  The  table  of  threes,  "the  lit- 
tle sisters  of  the  nines",  as  was  said  by  a  child,  has  special  applications  in  drama- 
tizing, in  the  game  of  "Threes  Out"  and  in  measurements  with  feet  and  yards. 
Measure  things  easily  accessible.  Finding  heights  of  pupils  is  an  interesting  ex- 
ercise. The  buying  and  selling  of  ribbon  (paper)  at  5  or  2  or  9  or  3  cents  a  yard 
furnishes  many  problems.  In  connection  with  the  tables  of  eights  and  that  of 
fours  much  work  is  done  with  circles.  Each  child  has  his  circle ,  made  of  paper 
or  pasteboard  and  at  least  six  inches  in  diameter.  This  he  folds  or  cuts  into  parts, 
as  halves,  fourths  and  eighths.  The  pupils  learn  to  draw  circles  and  to  bisect  and 
otherwise  divide  them.  They  find  out  by  inspection  how  many  fourths  or  eighths 
it  takes  to  make  a  whole.  These  sectors  and  the  symmetrical  parts  of  the  octagon, 
which  the  pupils  are  taught  to  construct  and  divide,  give  many  useful  illustrations 
of  the  tables  of  eights  and  that  of  fours.  "16  ounces  in  a  pound"  is  given  with 
the  eight  table  and  "4  quarts  in  a  gallon,"  shown  by  use  of  the  measures,  is  used 
in  the  number  stories  about  the  fours.  The  relative  lengths  of  half  notes,  quarter 
notes  and  eighth  notes  should  be  shown.  While  the  table  of  sevens  is  being  mem- 
orized the  fact  that  seven  days  make  a  week  gives  many  problems.  Besides  the 
usual  work  with  rectangles,  fractional  parts,  etc.,  there  is  much  buying  or  selling 
at  7  cents  a  yard,  pound  or  gallon.  In  learning  the  table  of  sixes  the  hexagon,  the 
six-pointed  star,  and  their  symmetrical  parts,  all  of  which  the  children  construct 
and  use  in  many  decorative  forms,  furnish  special  applications  of  the  table. 


30  ARITHMETIC 

Fourth  Work  in  the  elementary  state  textbook  is  begun  and  one-half  of  each 

Year         period  is  devoted  to  written  advance  in  the  book. 

Each  table  is  reviewed  with  new  applications,  such  as  factoring,  common 
multiples,  small  fractions,  etc.  The  battle  game  which  has  been  played  in  the 
learning  of  the  tables  is  now  used  so  as  to  correlate  them  and  to  give  much  rapid 
work  in  factoring  and  cancelation. 

The  word  "ratio"  is  used  interchangeably  with  "parts"  until  the  meaning  is 
clear.  Then  the  children  are  told  that  "ratio"  also  means  "times".  Much  drill 
is  given  upon  the  reciprocal  ratios  of  numbers. 

Long  division  is  begun  with  divisors  ii,  12,  no,  loi,  120,  121,  etc.  After 
a  review  of  the  mtultiplication  table  of  nines,  divisors  such  as  91,  910,  911,  89,  891, 
etc.,  are  used.  Each  table  is  reviewed  and  used  in  connection  with  long  division. 
In  both  short  and  long  division  each  figure  of  the  quotient  should  be  placed  direct- 
ly above  the  righthand  figure  of  that  part  of  the  dividend  which  produced  it. 

In  the  number  stories  stress  is  laid  upon  gain  and  loss  in  small  commercial 
dealings  with  which  children  are  familiar.  Simple  work  is  given  in  U.  S.  money, 
with  bills  and  accounts,  and  in  surface  and  boundary  measurements. 

Fifth  The  work  of  the  preceding  year  has  been  largely  anticipatory  to  the  sub- 

Year  jects  as  presented  in  the  latter  half  of  the  elementary  textbook,  and  in 
the  fifth  year  the  order  of  presentation  in  the  book  is  generally  followed.  Games, 
tests  and  general  devices  used  in  the  previous  grades  are  adapted  for  use  in  this 
grade.  In  connection  with  ratio  and  proportion  many  exercises  in  working  to  a 
scale  are  drawn  from  the  departments  of  Manual  Training  and  Domestic  Science. 

In  the  number  stories  there  is  much  simple  reasoning  based  upon  the  child's 
intuitions  of  number  and  upon  his  understanding  of  simple  business  affairs  as  in- 
terest, profit  and  loss,  etc.  Continued  number  stories  (problems  involving  two 
steps)  are  given.  Those  processes  of  written  work,  the  reason  for  which  the 
child  can  be  led  to  discover,  are  accounted  for,  as  for  instance,  the  addition  and 
subtraction  of  fractions.  Others,  as  the  division  of  one  fraction  by  another,  are 
given  simply  as  processes  leading  to  desired  results,  no  attempt  being  made  to 
force  knowledge  of  the  underlying  principles  into  the  immature  mind.  Formal 
analyses,  as  given  in  model  solutions,  are  shown  as  fine,  logical  ways  of  reason- 
ing, to  be  carefully  considered  but  not  memorized  without  understanding,  nor 
used  in  vain  repetitions  as  a  substitute  for  original  thinking. 

My  dear  students,  the  working  out  of  these  plans  which  I  have  merely 
sketched,  the  adjusting  of  the  work  to  the  abilities  of  individual  children,  the  care- 
ful preparation  in  class  for  indivdual  advance,  the  promoting  of  the  self-activity 
of  the  pupils,  the  insistence  upon  accuracy  and  the  preventing  of  mistakes,  the 
systematic  mastery  of  certain  portions  of  a  subject  before  others  are  attempted, — 
all  these  require  on  your  part  earnestness  in  the  work  and  a  determination  to  suc- 
ceed in  it  that  will  cause  you  to  give  much  energy  and  to  spend  many  hours  in 
serious,  fruitful  thinking  about  your  pupils'  development.  While  grasping  the 
general  plan  of  this  elementary  instruction,  if  you  concentrate  upon  the  little  span 
of  work  allotted  to  you,  laboring  intelligently,  faithfully,  and  lovingly  with  the 
little  ones  entrusted  to  your  care,  you  will  have  great  rewards.  Among  these  are 
the  pleasure  of  seeing  the  success  and  happiness  of  your  pupils,  and  also  the 
consciousness  of  your  own  growing  powers  as  a  teacher.  Hoping  that  these 
and  other  rewards  may  be  yours,  I  am 

Sincerely  your  friend  and  co-worker, 

AdElia  R.  Hornbrook. 

San  Jose,  Cal.,  Sept.,  1914. 


IV. 

DRILLS 


COMBINATION  CHART 

No.  «f 
New 
Comb. 

Ones 

1    2 
1    1 

3   4 
1    i 

5 
1 

6 
1 

7 
1 

8 
1 

J(   9 

Twos        -        -        - 

1  2 

2  2 

3   4 
2   2 

5 
2 

6 
2 

7 
2 

8 
2 

9i   ft 
2(   ° 

Groups  making  Ten 

- 

- 

9 
1 

8 
2 

7 
3 

6 

4 

II » 

Fives 

1    2 
5    5 

3   4 
5    5 

5 
5 

6 

5 

7 
5 

8 
5 

lie 

Groups  making  Five 

- 

■ 

- 

- 

4 
1 

!!• 

Groups  making  Eleven 

- 

-' 

10 
1 

9 
2 

8 
3 

7 
4 

§1^ 

Nines 

1    2 
9   9 

3    4 
2   9 

5 
9 

6 
9 

7 
9 

8 
9 

lie 

Groups  making  Nine 

- 

- 

8 
1 

7 
2 

6 
3 

t!» 

Doubles 

1    2 
1    2 

3   4 
3   4 

5 
5 

6 
6 

7 

I. 

8 
8 

SSs 

Eights 

1    2 
8   8 

3   4 

8    8 

6 
8 

6 
8 

7 
8 

8 
8 

ii3 

Groups  making  Eight 

- 

- 

7 
1 

6 
2 

5 
3 

t!» 

Other  New  Groups 

- 

- 

- 

- 

4 
3 

7'-    9 
6i    -^ 

Total 

- 

45 

31 


32 


ARITHMETIC 


LAWS  OF  COUNTING. 
Counting  by  Twos 

1.  When  the  counting-  begins  with 
two  or  any  even  number,  the  even 
numbers  are  named  in   order:   2,  4,   6, 

8,  10,  12,  14,  16,  18,  20,  etc. 

2.'  When  the  counting  begins  with 
one  or  any  odd  number,  the  odd  num- 
bers  are   named   in   order:    1,   3,   5,   7, 

9,  11,  13,  15,  17,  19,  21,  etc. 

Counting  by  Tens 

The  unit  figure  is  unchanged  and 
the  tens  figure  increases  one  each  time : 
11,  21,  31,  41,  etc. 

Counting  by  Fives 

The  alternate  unit  figures  are  tha 
same:  1,  6,  11,  16,  21,  26,  etc. 

Counting  by  Elevens 

Count  forward  ten  and  one ;  ] ,  32, 
23,  34,  45,  etc.  Note  that  the  unit  and 
tens  figures  are  each  increased  one  ex- 
cept when  the  unit  figure  is  nine. 

Counting  by  Nines 

Count  forward  ten  and  backward 
one:  7,  16,  25,  34,  43,  etc.  Note  that 
the  unit  figure  decreases  one,  and  the 
tens  figure  increases  one  each  time  ex- 
cept when  the  unit  figure  is  0.  Be- 
fore counting  by  nines  practice  count- 
ing backward  by  ones  from  9  to  0,  till 
it  can  be  done  without  effort  or  mis- 
take. 

Counting  by  Eights 

Count  forward  ten  and  backward 
two:  9,  17,  25,  33,  41,  49,  etc.,  and 
8,  16,  24,  32,  40,  48,  etc.  Note  that 
when  the  counting  begins  with  an  even 
number,  the  even  numbers  are  named  in 
reverse  order  in  the  units  place,  and 
when  it  begins  with  an  odd  number, 
the  odd  numbers  are  repeated  in  re- 
verse order.  Before  taking  up  cotmt- 
ing  by  eights,  practice  counting  back- 
ward by  twos  from  8  to  0  and  from 
9  to  1. 

Laws  similar  to  those  given  above 
govern  counting  by  twelves,  thirteens, 
nineteens,  twenties,  twenty-ones,  etc. 
The  pupil  should  be  encouraged  the 
not  required  to  discover  and  use  them. 


FIXING    THE    ADDITION-SUBTRAC- 
TION  COMBINATIONS. 

The  addition-subtraction  combina- 
tions should  be  well  learned  before  the 
pupil  is  required  to  use  them  miscel- 
laneously. This  requires  much  repeti- 
tion, and  demands  tact  and  patient 
perseverance  on  the  part  of  the  teach- 
er. The  following  exercises  will  be 
found   helpful : 

1.    Counting  Exercise 

This  exercise  is  a  kind  of  exposure, 
and  may  or  may  not  result  in  the  mas- 
tery of  a  given  set  of  combinations. 
With  the  ones,  twos,  and  tens  it  will 
usually   be    found   sufficient. 


2.  Column   Addition 

At  first  the  column  should  contain 
only  the  particular  number  under  con- 
sideration, except  at  the  foot,  where 
any  desired  number  may  be  placed. 
Later  any  whose  combinations  have 
been  previously  learned  may  be  placed 
in  the  column.  See  that  results  only 
are  named. 

3.  Decade  Drill 

For  example: 

2  2      2      2      2 

3  13     23     33    43,    etc., 


then: 


3      3       3 

2     12    22     etc. 


In  this  work  have  the  numbers  and 
the  results  named:  3  and  2  are  5,  13 
and  2  are  15,  etc.  Make  use  of  the 
number  chart :  Name  a  number  2 
greater  than  6,  16,  36,  etc. ;  2  less  than 
37,  17,  67,  etc. 

4.    Flash  Drill 

For  this  purpose  the  teacher  should 
have  a  set  of  cards  containing  the  for- 
ty-five combinations  and  blanks. 

Hold  a  card  before  the  class  for  a 
m.oment,  then  require  some  pupil  to 
eive  the  result. 


DRILLS 


33 


OUTLINE     FOR     DRILL     WORK     FOR 
SECOND   AND   THIRD   YEARS. 

The  ones  in  addition  and  subtrac- 
tion will  almost  certainly  be  learned 
thru  the  counting  exercise.  Test  and 
make  sure  by  drill  exercises. 

1.  Count  by  twos  beginning  with 
two  and  one ;  then  beginning  with  any 
number. 

2.  Add  single  columns  of  numbers 
consisting  of  twos  with  any  number 
at  the  bottom : 

2 

2  2 

2     2  2 

2     2     2  2 

2     2     2     2  2 

2     2     2     2     2  2 

7     7     7     7     7  7 


Observe  that  each  successive  column 
repeats  all  the  preceding  ones.  Later 
put  ones  among  the  twos  and  add. 
When  1  stands  above  another  1,  two 
should  be  added. 

3.     Supply       omissions  involving 
twos: 

(   )         4          2          5  (   ) 

2         ()()()  2 

9  6         11         7         10 

Give  results  as  follows :  2  and  7  are 
9;  4  and  2  are  6 ;  2  and  9  are  11,  etc. 

Later  put  the  work  in  regular  form 
for    subtraction : 

5  8  11 

_2        —2  —9 

Read  2  and  3  are  5 ;  2  and  6  are  8 ; 
9  and  2  are  11. 

4.  Count  by  tens  beginning  with 
10.     Then  beginning  with  1,  2,  3,  etc. 

5.  Name  a  number  10  more  than  a 
given  number,  10  smaller;  e.  g.,  name 
a  number  ten  more  than  75,  43,  28, 
etc.  10  less  than  96,  54,  87,  etc.  Use 
the   number  chart, 

6.  Learn  the  number  groups  mak- 
ing 10: 

9       8      7       6       5 
12       3      4      5 
Use    these    groups    in    columns    placing 


any  number  at  the  bottom.     Later  put 
twos  and  ones  in  the  columns. 
Supply   omissions : 

(   )         7  8         (   )       (   ) 

6         (   )       (   )         9  5 

10        10        10        10        10 
Use  flash  cards.     Keep  the  combina- 
tions   learned   on   the    board. 

7.  Count  by  fives  beginning  with 
5.  Then  beginning  with  1,  2,  3,  etc. 
Note  the  law  and  compare  it  with 
counting  by  tens. 

8.  Add  columns  consisting  of  fives 
with  any  number  at  the  bottom.  Later 
mix  in  ones,  twos,  and  groups  making 
10. 

9.  Learn  the  groups  making  five. 
Use  these  in  column  addition. 

10.  Supply  omissions  involving 
fives,  ones,  twos,  and  groups  making 
10.  Later  put  in  regular  subtraction 
form. 

11.  Count  by  twos  using  the  nu- 
meral frame  or  two  rows  of  objects. 

12.  Learn  the  doubles  to  9  and  9: 
123456789 
123456789 

13.  Make  use  of  decade  drills  to 
fix  these  and  other  combinations 
studied : 

6       6       6       6 

6     16     26     36     etc. 

14.  For  written  work  fill  the  blanks 
in  the  following: 


+5 

—5 

+2 

27 

41 

34 

16 

43 

59 

65 

88 

•92 

15.  Teach  the  multiplication  table 
of  the  ones,  twos,  tens,  and  fives  in 
the  order  named.  This  work  will  be 
comparatively  easy  for  the  children  for 
they  have  the  foundation  work  al- 
ready. The  ones  will  require  little 
if  any  drill.  Cards  containing  one 
combination  on  each  side  will  be  found 
helpful. 


34 


ARITHMETIC 


(1)  Count     by     twos     beginning 
wtih  two, 

(2)  Build    the    twos    in    multipli- 
cation  as   follows : 


(7)  Take  up  the  fives  in  a  sim- 
ilar manner.  Write  the  table  as 
shown : 


2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2     2 

2 

2 

2 

2 

2 

2 

2 

2    4     6     8  10  12  14  It)  18 

This  will  enable  the  pupil  to  get 
at  the  meaning  of  the  table.  Similar 
building  of  the  table  may  be  contin- 
ued in  the  tens  and  fives.  It  is  an 
unnecessary  burden  to  carry  it  fur- 
ther, for  by  that  time  the  child  will 
see  that  the  table  is  based  on  count- 
ing by  the  given  number  beginning 
with  the  number  itself. 

(3)     Write  the  twos  in  table  form ; 

1  X  2  =-  2 
2X2  =  4 
3X2  =  6 
4X2  =  8    etc. 

(4)  Drill  on  the  twos  using 
drills  and  forms  similar  to  those 
given  for  addition  and  subtrac- 
tion also  using  cards. 

(5)  Write  the  twos  in  divis- 
ion and  give  drills  as  suggested 
above. 

(6)  Give  exercises  with  the 
tens  similar  to  those  outlined  for 
the  twos.  After  the  table  has 
been  written  draw  a  line  as  here 
shown  and  have  the  pupils  note 
that  the  right  hand  digit  is  always 
0  and  that  the  others  are  the  same 
as  the  multiplier.  (Do  not  be  too 
ready  to  point  out  laws  to  the 
children.  Give  them  time  and  re- 
ward their  discoveries.)  : 


1  X  10  = 

1 

0 

2  X  10  = 

2 

0 

3  X  10  = 

3 

0 

4  X  10  = 

4 

0 

5  X  10  = 

5 

0 

1X5  = 

5 

2  X  5  =  10 

3  X  5  =  15 

4  X  5  =  20 

etc. 

rhen  separate  it  as  follows: 

2X5  =  1 

0 

3X5  = 

1 

5 

4X5  =  2 

0 

5X5  = 

2 

5 

6X5  =  3 

0 

7X5  = 

3 

5 

8X5  =  4 

0 

9X5  = 

4 

5 

10  X  5  =  5  I  0 

Have  the  pupils  note  that  when 
the  multiplier  is  an  even  number 
the  right  hand  digit  is  0  and  the 
other  is  half  the  even  number. 
Why  is  this?  When  the  multiplier 
is  an  odd  number  the  right  hand 
digit  is  5  and  the  other  is  half  the 
even  number  next  smaller  than 
the  multiplier.  For  other  drills,  see 
directions  given  under  multiplica- 
tion and  division. 

16.  Count  backward  by  ones 
from  9  to  0.  Repeat  the  exercises 
till  it  can  be  done  quickly  and  ac- 
curately, 

17.  Count  by  nines  beginning 
with  9,  1,  2,  3.  etc.  Note  the  la\v. 
Make  use  of  the  number  chart. 

18.  Teach  the  nines  in  addition 
and  subtraction.  Use  a  table,  and 
decade  drills. 

19.  Add  columns  consisting  of 
nines  with  any  number  at  the  bot- 
tom. Later  place  in  the  columns 
ones,  twos,  fives  and  groups  mak- 
ing one,  two,  ten,  or  five. 

20.  Supply  omissions  involving 
nines  and  other  combinations  learned. 
Also  use  the  regular  subtracting 
form.  Use  cards  containing  com- 
binations   studied    and    omissions. 

21.  Learn  the  groups  making  nine 

8       7       6       5 
12       3       4 

Use  these  in  column  work ;  fix  them 
by  decade  drills  and  other  exercises. 

22.  Count  by  elevens  beginning 
with  11,  1,  2,  S,  etc.  Note  the  law. 
Use  chart. 


DRILLS 


35 


23.     Learn     the     groups     making 
eleven : 

10      9       8      7       6 
12       3      4      5 


Use   these    in    column    work    and    in 
subtraction. 

24.  Count  backward  by  twos 
from  8  to  0  and  from  9  to  1.  Re- 
peat the  exercise  until  it  is  mastered. 

25.  Count  by  eights  beginning 
with  8,  1,  2,  3,' etc.  Note  the  law. 
Use  chart, 

26.  Learn  the  eights  in  addition. 
Use  the  chart,  decade  drills  and 
cards, 

27.  Add  columns  involving 
eights.  Also  supply  omissions  and 
subtract. 

28.  Review  the  groups  making 
eight : 

7       6       5      4 
12       3      4 

Use   them    in    column    work    and    in 
subtraction. 

4     7 

29.  Learn  the  groups,  3  6,  and 
fix  them  by  the  necessary  drills, 

30.  Review  the  forty-five  combin- 
ations by  groups.  (See  Group 
Chart.)  Encourage  the  addition  of 
two  figures  at  a  time  in  the  group 
work.  Where  the  sum  of  the  two 
is  more  than  ten  refer  it  to  ten  or 
twentv.     Thus : 

6 


13,  is  three  more 
than  10.  and  the  result  when  6  and 
7  are  added  together  in  a  column 
is  three  more  than  the  next  higher 
decade.     When 

9 

9 

18,  is  added  the  re- 
sult is  two  less  than  the  second 
next  higher  decade. 

Along  with  this  drill  work  there 
should  be  woven  in  much  concrete 
work.      The    problems    should    arise 


as  far  as  possible  from  the  pupils' 
employment  at  school,  at  home,  at 
play.  The  actual  weights  and  meas- 
ures in  the  hands  of  the  pupils  will 
furnish  material  for  problems, 
which  should  always  be  related  to 
the  neighborhood  occupations  and 
interests. 

MULTIPLICATION     AND     DIVISION. 

The  ones,  twos,  tens  and  fives 
have  already  been  studied.  They 
should  now  be  reviewed. 

1,  Count  by  elevens  beginning 
with  11.  Write  the  table  of  elevens 
in  multiplication.  Fix  this  by  drills. 
The  simple  law  and  the  rhythm 
make  the  learning  of  the  elevens 
easy, 

2,  Count  by  nines  beginning 
with  9, 

3,  Write  the  nines  in  Multiplica- 
tion : 

1X9=      |9 

2  X  9  =  1  |8 

3  X  9  =  2|7 

4  X  9  =  3  I  6 
.5  X  9  =  4  I  5 
6X9  =  514 
7  X  9  =  6  j  3 
8X9  =  712 
9X9  =  811 

Draw  a  line  between  the  units  and 
tens  of  the  product.  Note  how  the 
figures  run  in  each  column.  The  tens 
figure  represents  a  number  one  less 
than  the  multiplier.  The  sum  of  the 
units  figure  and  tens  figure  is  nine. 
These  facts  will  aid  the  nupil  at  first 
in  recalling  the  nines.  He  must  fin- 
ally become  able  to  recall  a  product 
immediately. 

Of  the  forty-five  combinations  of 
the  multiplication  table  up  to  9  X 
9,  thirty  have  been  studied.  If  tak- 
en up  in  order  there  will  be  five  new 
combinations  in  the  threes,  four  in 
the  fours,  three  in  the  sixes,  two  in 
the  sevens,  and  one  in  the  eights. 
These  and  also  the  twelves  should 
now  be  learned. 

To  memorize  the  multiplicaition 
table  was  thought  too  great  a  task 
for   the    child    a    few    centuries    ago. 


36 


ARITHMETIC 


and  he  was  required  to  carry  a  box 
containing  strips  on  which  the  ones, 
twos,  threes,  etc.,  in  multipHcation 
were  written.  When  he  was  re- 
quired to  multiply  by  six,  for  in- 
stance, he  took  out  the  six-strip  and 
referring  to  it  found  the  products. 
It  is  not  surprising  then  that  children 
forget  their  tables,  hesitate,  and 
make  mistakes.  Much  drill  and 
much  time  are   necessary. 

Drill  Wark  for  Multiplication  and  Di- 
vision. 

1.  Have  the  children  write  their 
own  tables. 

2.  Have  these  tables  repeated 
orally  by  each  member  of  the  class, 
looking  at  the  table  first,  without 
looking  at  it  later. 

3.  Have  the  tables  written  in 
tabular  form  as  shown  below : 

123456789 
2  4  6  8  10  12  14  16  18 
8      6      9     12    15    18    21     24    27 

This  may  be  done  as  a  class  ex- 
ercise on  the  board  or  paper.  After 
the  upper  row  has  been  written  the 
other  rows  may  be  written  in  any 
desired  order.  The  teacher  may 
dictate  a  row  thus: — Write  two 
times  seven,  five  times  seven,  seven 
times  seven,  four  times  seven,  etc. 
The  upper  row  may  be  written  in 
any  order  and  the  class  required  to 
fill  in  the  other  rows  as  a  seat  exer- 
cise. 

4.  Take  a  given  number  and 
write  its  factors  in  sets  of  two.  Thus: 

24  =  6X4  =  3X8=  2X12 
Use    in    this    way    all  the   products 
embraced  in  the  multiplication  table. 

5.  Find  the  numbers  in  each  dec- 
ade that  a  given  number  will  di- 
vide. Take  8  for  example:  1  to 
10,  8  =  1X8;  11  to  20,  16  =  2X8; 
51  to  60,  56  =  7X8,  etc. 

6.  Find  the  nearest  number  not 
larger  than  a  given  number  which 
another  number  will  divide.  Take 
9  for  example:  43,  36  =  4  X  9; 
57,  54  =  6  X  9 ;  etc. 


7.  Write  the  numbers  from  0  to 
12  in  a  vertical  column  in  any  de- 
sired order.  Multiply  any  number 
up  to  12  by  each  of  these.  This 
may  be  given  as  a  seat  exercise: 

X  6      X  8      X  4 


3 

6 

8 

2 

5 

9 
12 

4 

7 
11 

1 

0 

Then  erase  the  left  hand  column, 
place  the  sign  of  division  before  one 
of  the  numbers  at  the  top,  and  re- 
store the  first  column.  Vary  the 
form  as  suggested  under  addition 
and   subtraction. 

8.  Use  cards  as  in  addition  and 
subtraction.  There  are  forty-five 
combinations  in  multiplication  to 
9X9,  and  eighty-one  in  division  to 
81  ~-  9,  hence  sixty-three  cards  will 
be  required. 

These  drills  should  be  given  fre- 
quently even  after  the  tables  are  sup- 
posed to  have  been  learned. 

For  the  work  of  the  first  three 
years  the  teacher  should  be  supplied 
with  a  set  of  addition-subtraction 
cards  (63),  a  set  of  multiplication- 
division  cards  (63),  a  good  primary 
arithmetic  containing  many  drill  exer- 
cises, suggestions  and  illustration  for 
concrete  problems.  The  school  should 
be  supplied  with  balance  scales; 
weights,  one  ounce  to  two  pounds; 
foot  ruler ;  yard-stick ;  pint,  quart, 
half-gallon  and  gallon  measures ; 
clock  face  with  movable  hands ;  sets 
of  cards  containing  addition-subtrac- 
tion exercises,  sufficient  to  supply 
each  member  of  a  class  with  a  card 
of  either  set;  large  number  chart 
containing  numbers  from  1  to  100 
written  in  columns  of  tens,  and 
quantities  of  inch  cubes  and  inch 
squares. 


DRILLS 


37 


GROUP  CHART 

2=1 

10  =  } 

8 
2 

7 
3 

6 
4 

5 
5 

'      ? 

11       9 
11       2 

8 
3 

7 
4 

6 
5 

4-? 

2 
2 

12-§ 

8 
4 

7 
5 

6 
6 

5-! 

3 
2 

13      J 

8 
5 

7 
6 

- 

'      ? 

4 
2 

3 
3 

14      ? 

8 
6 

7 

1 

'      ? 

5 
2 

4 
3 

15-6 

8 

7 

»      ,' 

6 
2 

5 
3 

4 
4 

16      § 

8 
8 

9      ? 

7 
2 

6 
3 

5 

4 

18 

17      S 

_9 
-9 

ADDITION    OF    TWO    OR    MORE 
COLUMNS. 

There  should  be  no  serious  diffi- 
culty in  teaching  addition.  A  little 
reflection  will  convince  the  pupil  that 
only  units  of  the  same  order  can  be 
added. 

Care  should  be  taken  that  results 
only  are  named,  and  that  when  the 
sum  of  any  column  is  10  or  more, 
the  tens  of  the  sum  are  combined 
with  the  first  number  added  in  the 
next  column.  Encourage  the  addi-. 
tion  of  two  numbers  at  a  time  in 
groups,  and  insist  upon  it  when  the 


group   sum   does  not  exceed   11. 
6234 
2857 
1975 
7125 


18191 

Say  10,  21 ;  4,  11,  19 ;  11,  21 ;  10, 
18. 

Should  the  pupil  experience  seri- 
ous difficulty,  let  each  column  be 
added  and  the  results  combined. 
Such  work  is  not  uncommon  in 
practice  when  long  columns  are  to 
be   added. 


38 


ARITHMETIC 


6234 
2857 
1975 
7125 


21 

units 

17 

tens 

20 

hundreds 

16 

thousands 

18191 
Addition    is    a    fatiguing    exercise, 
and  it  is  best  that  short  columns  be 
used  at  first.     Keep  the  Group  Chart 
before  the  class, 

SUBTRACTION. 

Subtraction  is  the  finding  of  one 
addend  in  addition  when  the  sum 
and  the  other  addend  are  given,  and 
'should  be  approached  thru  supply- 
ing omissions  in  addition : 

26354  (  ) 

42641  42641 


68995 

68995 

Say  1    and 

4    are 

5 

4    and 

5    are 

9 

6    and 

3    are 

9 

2    and 

6    are 

8 

8496 

8496 

5376 

13872 

13872 

Say     6    and 

6    are 

12 

10    and 

7    are 

17 

5    and 

3    are 

8 

8    and 

5    are 

13 

A  similar 

process 

mav 

be  followed 

when  there 

are  more  th 

lan  two  ad- 

dends : 

3748 

9452 

9452 

6187 

6187 

19387  19387 

Say     9    and    8    are    17 

6,   14   and   4   are   18 
6    and    7    are    13 
10,    16   and  3   are   19 
The  usual  form  may  then  be  used. 
8769 
4532 


Say  2    and  7  are  9 

3  and  3  are  6 
5    and  2  are  7 

4  and  4  are  8 

14369 
6548 


7821 


4237 


Say    8  and  1  are  9 

4  and  2  are  6 

5  and  8  are  13 
7  and  7  are  14 

It  is  well  that  all  should  subtract 
in  this  manner,  but  not  wise  to  com- 
pel those  who  subtract  by  a  different 
method  to  change.  The  teacher, 
however,  should  make  this  her  habit- 
ual method  of  subtracting.  It  is  not 
necessary  that  the  subtrahend  should 
be   written   below   the   minuend. 

Teach  the  business  way  of  mak- 
ing change.  For  example,  if  the 
sale  amounts  to  $1.65  and  $5.00  is 
given  in  payment,  the  salesman 
would  say  $1.75,  $2.00.  $3.00,  $4.00, 
$5.00,  laying  down  in  succession  10c, 
25c,  $1.00,  $1.00  and  $1.00.  The 
change  is  taken  from  the  cash  regis- 
ter in  the  same  order. 

MULTIPLICATION 

Multiplication  presents  no  serious 
difficulty  after  the  tables  have  been 
learned ;  and  these  must  be  mastered. 
If,  however,  a  pupil  is  required  to 
multiply  by  8  and  does  not  know 
the  eight  table,  have  him  write  it  on 
a  strip  of  paper  and  make  reference 
to  it  when  necessary.  See  drills  for 
fixing  the   multiplication   table. 

Use  one  figure  as  a  multiplier  at 
first,  and  place  the  emohasis  on  the 
how  rather  than  the  why. 

The  main  difficulty  in  multiplying 
large  numbers  arises  from  the  fact 
that  children  foreet  the  tables.  To 
aid  pupils  in  recalling  and  fixing  the 
multiplication  table  use  the  same 
multiplier  till  it  is  well  learned, 
placing  the  emphasis  on  the .  tables, 
probably  the  6's.  7's.  and  8's,  that 
are  most  difficult.  The  same  sug- 
gestions apply  in  division  using  small 
divisors. 


DRILLS 


39 


DIVISION. 

Teach  long  division  first,  using 
divisors  not  exceeding  twelve.  See 
that  the  quotient  is  written  above  the 
dividend,  each  quotient  figure  being 
placed  directly  over  the  right  hand 
figure  of  the  dividend  used  in  ob- 
taining it.  First  multiply  a  number, 
and  then  reverse  the  process.  For 
example : 

3647 


349 


856)298744 
2568 


56 
32 
48 
24 

29176 
3647 

8  )  29176 
24 

51 

48 

37 
32 

56 
56 

After  the  pupil  has  become  able 
to  divide  by  long  division  using 
small  divisors,  introduce  short  divis- 
ion as  an  abbreviation.  Then  require 
that  short  divisiop  shall  be  used  for 
all  divisors  less  than  twelve. 
3647 


8  )  29176  Do  not  permit 
the  pupil  to  write  the  remainders,  or 
to  see  any  one  else  write  them. 

Long    Division    With    Large    Divisors. 

Long  division  when  the  divisors 
are  large  presents  three  difficulties  to 
the  beginner:  the  form,  the  approx- 
imation, and  the  testing.  These 
should  be  overcome  one  at  a  time. 
The  form  should  be  mastered  while 
using  small  divisors  as  suggested 
above. 

Divide  298744  by  856  and  then 
note  the  processes  by  which  the  re- 
sult is  obtained: 


4194 
3424 

7704 
7704 


First  we  say  8  in  29  three  times. 
Then  we  try  3  for  a  quotient  figure, 
and  find  that  after  dividing  29  by 
8  we  have  5  as  a  remainder,  which 
placed  with  the  8  makes  58,  a  num- 
ber more  than  sufficient  to  contain 
5  three  times.  Hence  the  first  quo*- 
tient  figure  is  3  and  the  next  divi- 
dend to  be  used  is  found  to  be  4194. 
8  in  41  five  times  with  a  remainder 
of  1,  which  being  joined  with  nine 
gives  19,  a  number  which  will  not 
contain  "  5  five  times.  Hence  the 
next  quotient  figure  is  not  5  but  4. 
So  we  continue  to  approximate  and 
test,  using  first  the  left  hand  figure 
then  testing,  using  one  or  more  of 
the  succeeding  figures  of  the  divisor. 

To  master  this  approximation  re- 
quires much  experience  on  the  part 
of  the  child  and  time  for  growth.  A 
rule,  however  explicit,  will  not  suf- 
fice. 

1.  Use  such  divisors  as  20,  30, 
40,  50,  60,  etc.  Then  the  left  hand 
figure  may  be  used  as  a  trial  divisor 
and  no  testing  is  needed. 

2.  Use  divisors  in  which  the  sec- 
ond figure  is  one:  as  41,  61,  713, 
816,  etc.  The  left  hand  figure  is 
used  as  a  trial  divisor,  and  the  result 
will  nearly  always  be  correct. 

Place  a  number  of  dividends  on 
the  board  and  have  pupils  find  and 
test  quotient  figures  by  inspection, 
using  divisors  like  21.  31,  41,  etc. 
Repeat  the  exercises  from  day  to 
day. 

3.  Use  divisors  in  which  the  sec- 
ond figure  is  2.  When  the  left  hand 
figure  is  used  as  a  trial  divisor  the 
result  will  need  more  careful  testing 
and  correcting. 

Place     several     dividends     on     the 


40 


ARITHMETIC 


board,  and  find  and  test  the  quo- 
tient figure  using  22,  32,  427,  521, 
etc.,  as  divisors. 


)68  )84  )122  )158  )198 
4.  Use  divisors  in  which  the  sec- 
ond figure  is  9.  The  number  next 
larger  than  the  left  hand  may  then 
be  used  as  a  trial  divisor  and  the  re- 
sult  tested   and  corrected. 

Continue  the  placing  of  dividends 
on   the  board   for  inspection  tests. 

In  this  way  the  pupil  will  gradu- 
ally learn  how  to  find  and  test  the 
quotient  figure.  The  teacher,  how- 
ever, must  be  satisfied  with  slow 
progress,  must  persevere,  and  be 
patient. 

FRACTIONS. 

Children  will  almost  certainly 
know  something  of  fractions  when 
they  enter  school.  What  child  has 
not  had  to  share  his  apple,  his  or- 
ange, or  his  marbles.  There  can  be 
no  valid  objection  against  the  intro- 
duction of  fractions  with  small  de- 
nominators in  the  early  years  of  the 
course.  These  fractions  should  be 
concrete:  as  3  fourths  of  a  gallon, 
2  thirds  of  a  foot,  etc.,  and  the  de- 
nominator should  be  expressed  in 
words,  and  thought  of  as  the  names 
of  the  parts.  Similar  fractions  may 
be  added  or  subtracted,  or  a  frac- 
tion may  be  multiplied  or  divided  by 
a  whole  number.  No  rules  should 
be  given.  A  more  extended  study  of 
fractions  should  be  taken  up  in  the 
fifth  year,  and  a  fuller  treatment  be 
given  during  the  sixth  year  and 
later. 

KEDUCTION. 

1.  Reducing  improper  fractions 
to  whole  or  mixed  numbers  and 
whole  or  mixed  numbers  to  improp- 
er fractions  presents  no  serious  dif- 
ficulty. 

2.  Reducing  to  higher  terms  may 
be  illustrated  by  taking  two  halves 
of  an  apple  and  dividing  each  into 
2  or  3  parts.  It  may  then  be  seen 
that  %  =  %  =  %,  etc 


A  better  way,  however,  is  to  fur- 
nish the  pupils  with  strips  of  paper 
of  uniform  length  and  have  these 
folden   as   follows : 

a.  Lay  one  strip  down  without 
folding. 

b.  Fold  one  strip  and  crease  it 
in  the  center. 

c.  Fold  a  third  strip  and  crease 
it  into   fourths. 

d.  Fold     a     fourth     strip     and 
crease    it    into    eighths. 

Place  these  strips  side  by  side, 
and  it  will  readily  be  seen  that 

y2  =  %  =  %;.   %  =  %;  %  =  % 

e.  Draw  diagrams  on  the  board 
representing   these    divisions. 


In  a  similar  manner  diagrams  may 
be  drawn  representing  the  relations 
of  halves,  thirds,  and  sixths ;  of 
halves.  thirds,  fourths,  sixths, 
twelfths,  etc. 

The  divided  apple  is  concrete. 
The  creased  paper  or  the  drawing  is 
a  representative  concrete  and  be- 
comes a  tool  which  the  pupils  may 
use  or  image  in  determining  other 
fractional  relations.  Dealing  with 
fifths,  for  instance,  it  will  be  found 
that 

%  =  yio  =  %o  =  '%5,  etc. 

Observing  these  and  other  frac- 
tional equivalents,  which  may  be 
worked  out,  the  pupil  will  get  and 
understand  the  rule.  He  win  also 
see  that  fifths  cannot  be  reduced  to 
ninths,   fourths  to   elevenths,   etc. 

3.  Reducing  fractions  to  lower 
terms  follows  as  a  corollary  from 
number  two  and  needs  no  illustra- 
tion. 

4.  Reducing  fractions  to  a  com- 
mon denominator  should  be  first  in- 
troduced in  connection  with  addition 
and  subtraction.  It  should  later  be 
studied   as   a   separate   topic.      Small 


DRILLS 


41 


fractions  should  be  given  as  a  rule 
and  the  work  done  principally  by  in- 
spection. If,  because  of  the  large- 
ness of  the  c6mmon  denominator, 
written  calculations  are  required,  let 
these  be  done  as  side  work.  The  least 
common  denominator  is  the  1.  c.  m. 
of  the  denominators  and  should  be 
found  as  set  forth  under  that  topic. 

ADDITION    AND    SUBTRACTION 

There  is  no  special  difficulty  in 
teaching  addition  and  subtraction  of 
fractions,  but  care  should  be  taken 
that  good  forms  are  used. 

1.  Give  much  drill  in  adding  and 
subtracting  similar  fractions.  This 
work  may  be  introduced  along  with 
addition  and  subtraction  of  whole 
numbers  provided  the  denominator 
is  expressed  in  words.  Write  frac- 
tions to  be  added  or  subtracted  un- 
der each  other,  especially  when 
there  are  mixed  numbers. 


16 


36 


63/4     = 

-    6  9^2 

71/0     = 

-    7  %2 

91/3     = 

■-   9  4/12 

12%   = 

=  12i%2 

36%o  = 

342%2 

('%2 

=  2%2) 

2.  Use  full  form  for  work  in  the 
fifth  year  writing  the  reduced  an- 
swer directly  under  the  first  column. 
Use  a  similar  form  in  subtraction. 
Write  out  in  full  when  the  fraction- 
al part  of  the  subtrahend  exceeds 
the  fractional  part  of  the  minuend. 
Avoid  the  term  '"borrow."  Take  one 
from  the  52  and  change  it  to 
twenty-fourths. 

27%  =  2720/^4 
19%  =  19  %4 


811/24 

52%  =  5215/4  =  5139/04 

272/3=2716/4= 271 6/;4 

242^24 

3.  Use  the  abbreviated  form 
when  addition  and  subtraction  are 
studied  in  the  sixth  and  subsequent 
years. 


121/4  1 

4 

83  %2     ! 

21 

27%     1 

6 

36  %       j 

8 

63%6  1 

7 

1 

103^6  ii%6=lK6  47i%6  |^%« 
Add  and  reduce  the  fraction  first, 
place  the  fraction  of  the  result  under 
the  fraction  of  the  addends,  and  add 
the  integral  part,  if  any,  along  with 
the  other  whole  numbers. 

ALIQUOT  PARTS. 

The  customary  way  of  presenting 
aliquot  parts  is  to  give  a  list  of 
aliquot  parts  of  100  and  require 
that  they  shall  be  memorized,  then 
to  give  rules  for  the  use  of  these 
numbers  in  multiplication  and  divis- 
ion. No  effort  is  made  to  assist  the 
pupil  in  memorizing  the  list,  or  to 
lead  him  to  understand  and  appreci- 
ate the  principle  underlying  the 
rules.  Such  work  has  no  education- 
al value  and  will  benefit  only  the 
pupil  who  soon  engages  in  some 
commercial  business  which  makes  use 
of  it. 

In  what  follows  an  effort  is  made 
to  relate  the  work  so  as  to  aid  the 
memory  and  so  that  laws  for  multi- 
plication and  division  will  be  im- 
pressed. 

1.  Count  by  2^  beginning  with 
2^.  Continue  the  counting  till  a 
law  is  found.  21/,  5,  71/,  10,  etc. 
Note  that  the  ending  are  repeated 
in  order  in  sets  of  four.  Note  also 
that  4  X  23^  =  10,  8  X  2^4  =  20 ; 
in  general  that  there  will  be  as 
many  tens  in  the  product  as  there 
are  fours  in  the  multiplier.  Con- 
versely, when  10,  20,  30,  etc.,  is  di- 
vided by  21/^  the  quotient  will  be  as 
many  times  4  as  the  number  has 
tens. 

Multiply  21/2  by  12,  28,  24,  36^ 
16,  32,  44,  etc. 

Divide  40,  70,  30,  50,  90,  60, 
80  by  2><. 

2.  Write  and  learn  the  table  of 
214  to  4  X  21/.  If  any  number  is 
multiplied  by  2^^,  the  product  will 
contain  as  many  tens  as  the  number 
has  fours  and  the  units  will  be  25<2, 


42 


ARITHMETIC 


5,  or  7^,  according  as  there  is  a 
remainder  of  1,  2,  or  3.  Thus  25  = 
6X4  +  1,  hence  25  X  23^  =  62>^  ; 
35  =  8  X  4  +  3,  hence  35  X  2>^  = 
87K. 

Any  number  which  ends  in  0,  2^, 
5,  or  7^  is  exactly  divisible  by  2^. 
To  obtain  the  quotient,  multiply  the 
tens  (all  above  units)  by  4  and  to 
the  product  add  the  quotient  ob- 
tained by  dividing  the,  units  by  2j^. 
57>4  -^  25^  =  5  X  4  +  3  =  23; 
115  -f-  2j^  =  11  X  4  +  2  =  46. 

3.  Deal  with  Sys  in  a  similar 
manner  and  obtain  laws  for  multi- 
plication and  division  by  3^.  Give 
much  exercise  in  the  use  of  these 
laws. 

4.  Count  by  25;  learn  the  twen- 
ty-five table  in  multiplication  to 
4X25.  Multiply  and  divide  by  25 
until  such  work  becomes  easy.  Note 
the  similarity  of  the  25  and  2^ 
tables. 

5.  Count  by  Uy,  to  100.  Place 
the  results  side  by  side  with  the  re- 
sults obtained  by  counting  by  2J/2 
and  25. 

12>^ 
2y2        25  25 

5  50  50 

62y2 

iy2        75  75 

87>^ 
10      ,    100  100 

Note  that  the  unit  figures  agree  with 
the  units  of  the  2^/2  series ;  also  that 
every  second  result  agrees  with  a 
result  in  the  25  series. 

6.  Write  the  12>^  table  in  multi- 


plication to  8  X  12>^.  Place  the 
table  aside   for  quick  reference. 

If  a  number  is  multiplied  by  12^^, 
the  product  will  contain  as  many 
hundreds  as  the  number  has  eighths 
and  the  tens  and  units  of  the  prod- 
uct will  be  the  product  of  12%  and 
the  remainder  as  shown  in  the  table. 
43  =  5  X  8  +  3,  hence  43  X  121/2  = 
5371/9  ;  77  =  9  X  8  +  5,  hence  77  X 
121/2  =  9621/2. 

A  number  which  ends  in  any  num- 
ber found  in  the  table  is  exactly  di- 
visible by  1214.  The  quotient  is 
eight  times  the  hundreds  figure  (all 
over  tens)  plus  the  quotient  ob- 
tained by  dividing  the  units  and  tens 
by  121/2.  6371/  -- 12^  =  8X6  +  3 
=51 ;  II871/2  -=- 121/  ^8  X  11  +  7  = 
95. 

7.  Take  up  33>^,  16^,  and  8>^ 
in  succession  and  deal  with  them  in 
a  similar  manner,  referring  the  re- 
sults to  the  3%  series.  If  the  series 
are  placed  side  by  side,  the  similar- 
ity and  the  laws  will  be  emphasized. 
Select  the  new  results  in  each  series 
in  order  and  learn  them. 

In  this  connection  take  up  bills 
and  accounts.  Use  regular  business 
forms.  Show  the  class  samples  of 
bills  from  different  mercantile  es- 
tablishments. Have  the  pupils  rule 
their  own  bills  and  fill  them  out  in 
a  neat  and  orderly  manner.  En- 
courage getting  results  by  inspec- 
tion. Let  other  necessary  calcula- 
tions be  done  as  side  work.  Intrcv- 
duce  receipts,  drafts,  and  checks.  It 
will  be  found  that  this  touching  of 
actual  business  forms  and  customs 
will  be  very  attractive  to  the  pupils. 


V. 

COURSE 


FIFTH  YEAR. 
FRACTIONS. 

Impress  and  deepen  the  fraction 
concept  by  divided  apples,  oranges 
and  other  objects ;  ruler,  yard-stick, 
pint,  quart,  and  gallon  measures, 
pound  and  other  weights,  etc.,  and 
folded   paper. 


Change  whole  numbers  and  mixed 
numbers  to  fractions. 

Change  improper  fractions  to 
mixed  numbers. 

Add  and  subtract  fractions  having 
a  common  denominator.  Add  and 
subtract  mixed  numbers  in  a  similar 
manner. 


COURSE 


43 


Factor  products  found  in  the  mul- 
tiplication table. 

Find  by  inspection  the  common 
multiples  of  numbers.  The  common 
multiples  should  be  the  products 
found  in  the  multiplication  table. 
While  the  least  common  multiple 
should  be  sought,  it  is  not  vital  that 
these  terms  should  be  singled  out 
for  special  study. 

Find  by  inspection  common  di- 
visors of  numbers  to  twenty,  to 
thirty,  to  fifty.  The  numbers  should 
be  products  found  in  the  multiplica- 
tion table,  and  two  or  not  more  than 
three  numbers  should  be  used  at  a 
time. 

Reduce  fractions  to  larger  denom- 
inators, Use  objects,  folded  paper, 
and    drawings   to   illustrate. 

Add  and  subtract  fractions  of  dif- 
ferent denominators  using  the  full 
form.  The  common  denominator 
should  be  limited  to  products  found 
in   the   multiplication   table. 

Reduce  fractions  to  their  lowest 
terms. 

Use  cancellation  where  there  is  in- 
dicated multiplication  and  division. 

Multiply  a  fraction  by  a  whole 
number.  Multiply  a  mixed  number 
by  a  whole  number.  When  the  in- 
tegral part  is  small  the  expression 
should  be  reduced  to  an  improper 
fraction;  when  it  is  large  the  frac- 
tional part  should  be  multiplied  first. 
If  the  multiplier  does  not  exceed 
twelve,  which  should  be  the  rule  at 
this  time,  the  carrying  should  be 
done  at  once  as  in  other  multiplica- 
tion. 

Divide  a  fraction  by  a  whole  num- 
ber. Divide  a  mixed  number  by  a 
whole  number.  When  the  integral 
part  is  small  reduce  to  an  improper 
fraction ;  when  it  is  large  divide  as 
in  whole  numbers  and  reduce  only 
the  remainder  to  a  fraction.  Use 
small  divisors. 

Find  a  fractional  part  of  a  whole 
number.  Multiply  a  whole  number 
by  a  fraction.  Use  cancellation. 
Multiply  a  whole  number  by  a  mixed 
number.  Do  not  reduce  to  improper 
fractions  unless  the  integral  part  is 
quite  small. 


Find  a  fractional  part  of  a  frac- 
tion. Multiply  a  fraction  by  a  frac- 
tion. Give  rule  and  use  cancellation. 
Reduce  mixed  numbers  to  improper 
fractions. 

Find  a  number  when  a  fractional 
part  of  it  is  given.  Divide  a  whole 
number  by  a  fraction.  Indicate  the 
work    and    use    cancellation. 

Divide  a  fraction  by  a  fraction. 
Use  inversion,  indicate  the  work, 
and  cancel, 

DECIMALS. 

Introduce  decimals  by  using  prob- 
lems requiring  addition  and  subtrac- 
tion of  United  States  money.  Add 
and   subtract   other   decimals. 

Multiply  United  States  money  by 
a  whole  number,  and  by  a  mixed 
number  using  forms  for  bills  and 
other  accounts.  Multiply  other  dec- 
imals by  whole  or  mixed  numbers. 

Divide  United  States  money  by  a 
whole  number.  Divide  other  deci- 
mals by  a  whole  number.  Reduce 
common  fractions  to  decimals.  Place 
the  quotient  above  the  dividend,  and 
place  the  decimal  point  in  the  quo- 
tient as  soon  as  the  decimal  point  in 
the  dividend  is  reached. 

Multiply  a  decimal  by  a  decimal. 
Use  the  word  per  cent  and  the  char- 
acter, %,  interchangeably  with  hun- 
dredths. Find  any  per  cent  of  a  num- 
ber. Express  the  per  cent  decimally 
when  using  it.  Find  interest  on 
money,  confining  the  time  to  years, 
or  years  and  months  which  will 
make  easy  fractions  of  years. 

Divide  a  decimal  by  a  decimal.  First 
mark  off  as  many  decimal  places  in 
the  dividend  counting  from  the  decimal 
point,  as  there  are  decimal  places  in 
the  divisor.  Place  the  decimal  point 
in  the  quotient  as  soon  as  the  mark 
in  the  dividend  is  reached. 

SIXTH  YEAE. 

State   Advanced   Arithmetic. 

It  is  assumed  at  this  time  that  the 
pupil  has  an  elementary  knowledge 
of  fractions,  that  he  can  add,  sub- 
tract,   multiply    and    divide    decimals 


44 


ARITHMETIC 


and  that  he  has  done  some  simple 
work  in  percentage. 

Review  reading  and  writing  of 
whole  numbers  and  decimals  to  bil- 
lions, by  the  Hindu-Arabic  notation. 
Learn  to  count  to  one  hundred  by 
the  Roman  notation,  learn  the  sig- 
nificance of  I,  V,  X,  h,  C,  D,  and 
M.  Learn  the  additive  and  subtrac- 
tive  laws.  Express  various  dates  in 
the  Roman  notation. 

Recall  and  learn  the  tables  for 
long  measure  (inches,  feet  and 
yards),  liquid  measure  (gills,  pints, 
quarts  and  gallons.) 

Review  addition  and  subtraction 
of  whole  numbers  and  decimals.  In 
this  review  follow  the  group  chart, 
and  insist  that  two  figures  be  added 
at  once  when  their  sum  does  not  ex- 

2  4 
ceed  12.     For  example  when  3  or  i 

stand  one  above  the  other  for  addi- 
tion, add  5.  Count  by  10,  11,  9,  12, 
and  8,  emphasizing  the  law,  and  then 
add  groups  making  one  of  these 
numbers.  Give  examples  in  com- 
pound addition  and  subtraction  in- 
volving the  tables  learned. 

Review  multiplication  of  whole 
numbers  and  decimals.  Give  system- 
atic and  persistent  drills  on  the  mul- 
tiplication tables,  emphasizing  the  6's, 
7's  and  8's.  Impress  the  Iciws  for 
5's.  Give  exercises  in  multiplication 
of  compound  numbers.  Take  up 
special  cases  in  multiplication :  ( i ) 
Multiplying  by  one  with  any  num- 
ber of  ciphers  annexed;  (2)  Multi- 
plying by  any  number  with  ciphers 
annexed;  (3)  Multiplying  by  25  and 
I2>^  ;  (4)  Multiplying  by  33/3,  16^, 
Sy3  and  66%. 

Introduce  bills  and  accounts.  Use 
regular  bill  paper. 

Find  areas  of  rectangles  and  work 
out  table  for  square  measure  for 
square  inches,  square  feet,  and 
square  yards. 

Find  volume  of  rectangular  solids 
and  work  out  table  for  cubic  inches, 
cubic   feet,   and   cubic  yards. 

Review  division  of  whole  numbers 
and    decimals.      Use    short    division 


where  the  divisor  does  not  exceed 
twelve  and  place  quotient  above  or 
below  the  dividend  as  is  most  con- 
venient. Use  long  division  when 
the  divisor  exceeds  twelve.  When 
the  divisor  is  a  decimal  mark  off  as 
many  decimal  places  in  the  dividend 
as  there  are  decimal  places  in  the  di- 
visor. Place  the  decimal  point  in  the 
quotient  as  soon  as  the  mark  in  the 
dividend  is  reached. 

Reduce  a  denominate  number  to 
units  of  a  higher  order. 

Take  up  special  cases  of  division, 
such  as  (i)  Dividing  by  one  with 
ciphers  annexed;  (2)  Dividing,  by 
any  number  with  ciphers  annexed; 
(3)  Dividing  by  25  and  I2>^ ;  (4) 
Dividing  by  331^,   16%  and  8>S. 

Give  some  work  in  compound  di- 
vision using  such  divisors  as  2,  3, 
4,  5 ;  also  some  inverse  problems  in 
rectangles  and  rectangular  solids. 

FACTORING. 

Factor  (i)  Products  in  the  multi- 
plication table,  writing  two  factors 
and  then  resolving  these  factors  into 
prime  factors;  (2)  Other  numbers 
less  than  100;  (3)  Numbers  of  three 
figures  in  which  the  right  hand  fig- 
ure  is   o. 

Learn  and  apply  the  laws  for 
divisibility:  (i)  Two,  five  and  ten; 
(2)  Four  and  twenty-five;  (3)  Nine, 
three  and   six. 

COMMON  DIVISORS. 

Find  common  divisor  by  inspec- 
tion  of  numbers  under  fifty. 

Find  the  greatest  common  divisor 
of  given  numbers  under  two  hun- 
dred by   factoring  one  of   them. 

Apply  common  divisors  in  reduc- 
ing fractions  to  their  lowest  terms, 
and  in  cancellation. 

COMMON  MULTIPLES. 

Find  common  multiples  of  num- 
bers by  inspection,  when  such  com- 
mon multiples  do  not  exceed  fifty. 
Find  the  least  common  multiple  by 
factoring,  the  1.  c.  m.  not  to  exceed 
one  hundred. 


COURSE 


45 


ADDITION    AND    SUBTRACTION    OF 
FRACTIONS. 

Add  fractions  using  the  abbrevi- 
ated form,  the  common  denominator 
being  a  product  of  the  multiplication 
table;  subtract  fractions  using  the 
abbreviated  form,  the  common  de- 
nominator not  to  exceed  fifty 

MULTIPLICATION  AND  DIVISION  OF 
FRACTIONS. 

Multiply  a  fraction  by  a  whole 
number,  give  rule  and  have  it 
learned.  Multiply  a  mixed  number 
by  a  whole  number ;  multiply  a  com- 
plex decimal  by  a  whole  number. 
Reduce  denominate  fractions  to  in- 
tegers  of  lower  orders. 

Divide  a  fraction  by  a  whole  num- 
ber, (a)  when  the  numerator  ex- 
actly contains  the  divisor;  (b)  When 
the  numerator  does  not  exactly  con- 
tain the  divisor.  Give  rule  and  have 
it  learned.  Divide  a  mixed  number 
by  a  whole  number.  Divide  a  com- 
plex decimal  by  a  whole  number. 

Multiply  a  whole  number  by  a 
fraction,  also  by  mixed  number  and 
a  complex  decimal.  Multiply  a  frac- 
tion by  a  fraction,  a  mixed  number 
by  a  fraction,  a  complex  decimal  by 
a  fraction.  Similarly  multiply  by  a 
mixed  number  and  by  a  complex 
decimal. 

Divide  a  whole  number  by  a  frac- 
tion, a  fraction  by  a  fraction,  a  mixed 
number  by  a  fraction,  and  a  com- 
plex decimal  by  a  fraction.  Sim- 
ilarly divide  by  a  mixed  number  and 
a  complex  decimal. 

FRACTIONAL  RELATIONS. 

Find  a  required  part  of  a  number. 

Find  a  number  when  a  certain 
part  of  it  is  given. 

Find  what  part  one  number  is  of 
another. 

Apply  the  same  to  fractions. 

SEVENTH   YEAR. 

PERCENTAGE. 

Express  per  cent  as  a  decimal  and 
apply  the  same  in  finding  required 
per  cent  of  a  number. 


Express  certain  per  cents  as  com- 
mon fractions  and  apply  the  same  in 
finding  a  required  per  cent  of  a 
number  or  fraction. 

Find  a  required  number  when  a 
certain  per  cent  of  it  is  given,  first 
expressing  the  per  cent  decimally, 
or  as  a  fraction. 

Express  a  decimal  as  per  cent ; 
also  change  a  comm.on  fraction  to 
per  cent. 

Find  what  part  one  number  is  of 
another  and  find  whit  per  cent  one 
number   is   of  another. 

Find  a  number  a  given  part  great- 
er than  another  and  find  a  number  a 
given  per  cent  greater  than  another. 

Find  a  number  a  given  part  less 
than  another ;  also  a  given  per  cent 
less  than   another. 

Find  a  num.ber  when  another 
number  a  given  part  or  per  cent 
greater  than  it  is  given;  also  when 
one  given  part  or  per  cent  less  than 
it  is  given. 

APPnCATION  OF  PERCENTAGE. 
INTEREST. 

Find  interest  for  years,  years  and 
months,  and  months  and  days. 

Find  the  time  by  compound  sub- 
traction and  by  counting  the  ex- 
act number  of  days.  In  the  latter 
case,  reckon  360  days  as  a  year  for 
commercial  transaction,  and  365  for 
a  year  for  exact  interest. 

Study  commercial  paper,  including 
notes,  drafts,  checks  and  money  or- 
ders. 

Study  banking,  including  bank 
discount  and  savings  accounts. 

EIGHTH  YEAR. 

Build  up  squares  and  extract 
square  root.  Study  and  solve  right 
triangles. 

SURFACES. 

Land  measure,  including  United 
States  land  divisions,  land  surveyor's 
chain.     Vara, 

Lumber  measure,  including  shin- 
gling. 

Plastering,  papering  and  carpet- 
ing. 

Areas  of  parallelograms,  triangles, 
trapezoids  and  circles. 


46 


ARITHMETIC 


VOLUMES. 

Rectangular     solids, 
cylinders. 

Pyramids  and  cones. 
Frustums. 


prisms 


Spheres, 
and         Problems  in  analysis  and  ratio  and 
proportion. 

Longitude   and  Time,   including  a 
discussion  of  Standard  Time. 


VI. 

REVIEWS 


1.    READING    AND    WRITING    NUM- 
BERS. 

HINDU-ARABIC     NOTATION. 

1.  Numbers  are  usually  expressed 
in  writing  by  the  use  of  ten 
characters  called  digits.  These  digits 
are  i,  2,  3,  4,  5,  6,  7,  8,  9,  o,  and 
standing  alone  represent — one,  two, 
three,  four,  five,  six,  seven,  eight, 
nine,  and  naught,  respectively,  and 
are  so  named. 

In  writing  a  number  each  digit 
represents  a  value  dependent  on  the 
place  which  it  occupies,  this  value 
increasing  from  right  to  left  in  a 
ratio  tenfold.  In  expressing  a  whole 
number  the  right  hand  digit  repre- 
sents the  same  value  as  the  digit 
when  standing  alone ;  when  in  the 
second  place  counting  from  the  right 
it  represents  a  value  ten  times  as 
great;  when  in  the  third,  one  hun- 
dred times  as  great,   and   so  on. 

Thus,  in  11 11  the  right  hand  digit 
represents  one,  the  next  ten,  the  next 
one  hundred,  and  the  next  one 
thousand.  In  22222  the  right  hand 
digit  represents  two,  the  next  twenty, 
the  next  two  hundred,  the  next  two 
thousand,  and  the  next  twenty  thou- 
sand. What  does  each  digit  repre- 
sent in  163435? 

For  convenience  in  reading  these 
numbers  are  grouped  into  periods 
of  three  figures  each  beginning  with 
units,  or  the  right  hand  digit  in 
whole  numbers.  These  groups  are 
frequently  separated  by  commas  to 
aid   the   eye   in   distinguishing  them. 

The  first  five  periods  are  named 
as  follows  beginning  at  the  right: 
units,  thousands,  millions,  billions 
and  trillions.  Each  place  is  called 
an  order,  and  beginning  at  the  right 


the  orders  are  named  as  follows : 
units,  tens,  hundreds,  thousands, 
ten  -  thousands,  hundred  -  thousands, 
millions,  ten-mrllions,  hundred-mil- 
lions, etc. 

Separate  the  following  numbers 
into  periods  and  name  the  periods 
and  the  orders:  476328,  765342708, 
62840070903. 

It  should  be  noted  that  each  per- 
iod except  the  left  hand  one  must 
have  three  places  and  that  o  is 
written  in  each  vacant  order. 

2.  Reading  Numbers.  In  reading 
numbers  the  periods  are  read 
in  order  beginning  with  the  one 
at  the  left.  The  word  and  is  un- 
necessary in  reading  whole  numbers 
and  should  be  omitted.  For  ex- 
ample, 60,705,821,062  should  be  read 
sixty  billion,  seven  hundred  five 
million,  eight  hundred  twenty-one 
thousand,  sixty  -  two.  964,000,083 
is  read  nine  hundred  sixty-four  mil- 
lion,  eighty-three. 

Read  the  following  numbers: 

1.  671;  671,000;  671,000,000 

2.  403;  4,003;  40,300 

3.  520;  52,000;  5,200,000 

4.  89;  89,089;  8,900,890 

5.  47,407,047 ;    4,700,400,747 

6.  67,004 ;  149,820 
600,700,100. 

400,003,000. 
50,000,020. 
890,089,809. 
407,470,047,407. 
74,000,937,506. 
Name  the  order  and  give  the  value 
expressed    by    each    significant    digit 
in  the  following  numbers,  then  read 
'  each  number : 
682,251;       403,075,209;       20,704,641 


REVIEWS 


47 


3.  Writing  Numbers.  In  writing 
numbers  each  period  except  the  left 
one  must  be  full,  each  vacant  order 
being  filled  with  0. 

Place  the  following  on  the  paper 
or  board  and  write  the  numbers 
under  it: 

Trill.      Bill.       Mill.    Thous.     Units 
h.t.tr.  h.t.b.    h.t.m.  h.t.th.  h.t.u. 
000,000,000,000,000 

Write  the  following  numbers,  sep- 
arating the  periods  by  commas: 

1.  Two     thousand      six    hundred 

fifty-three. 

2.  Six   thousand     sixty-six. 

3.  Four      million       four      hundred 

thousand    four. 

4-  Twenty-seven  billion  two  hund- 
red seven  thousand  two 
hundred  seventy. 

5.  One    hundred     sixteen    trillion 

eighty-four  billion  five  hund- 
red sixty-one  million  four 
hundred. 

6.  Four   million    27   thousand     36. 

7.  Twelve   billion     305    thousand. 

8.  Three      hundred      seventy  -  one 

trillion  60  million  8  thou- 
sand    406. 

9.  200  billion    2  million    20  thou- 

sand   202. 

10.     ^2   million    7   thousand    40. 

Write  the  following,  omitting  the 
commas : 

1.  ^00    thousand    25. 

2.  902  million    3  thousand    800. 

3.  47  billion    706  million    9. 

4.  6  trillion    15  million    28. 

5.  391  billion    408  thousand. 

6.  560   trillion     38    billion     7    mil- 

lion    500   thousand. 

7.  74  million    200  thousand    65. 

8.  400   million     5   thousand    6. 

9.  7  trillion    5  million    3  thousand 

I. 

10.     60  billion    800  million    4  thou- 
sand   20. 
Read   all   the   numbers   v/hich   you 

have  written. 


FACTORING. 
FACTORING  BY  INSPECTION. 

Small  numbers  may  be  factored 
by  inspection  by  first  separating  each 
number  into  two  factors  and  factor- 
ing the  factors  when  possible. 

24=4X6=2X2X2X3 
54=6X9=2X3X3X3 
28=4X7=2X2X7 
In  like  manner   factor  the  follow- 
ing numbers : — 

1.  36,  42,  45,  32,  56,  ^i,  75,  81, 
44,  48.  It  is  not  necessary  that  the 
pairs  of  factors  be  written.  They  may 
be  thought  and  the  final  factors  only 
need  be  written. 

Factor  the  following  numbers, 
writing  the  prime   factors  onl)': — 

2.  64,  66,  84,  'd^,  108,  144,  88, 
99,  27,  125. 

Numbers   which   end   in   0  contain 
the  factor  10  or  2  X  5.     In  factor- 
ing such  numbers  the  2X5  should 
be  written  first,  thus: 
210=2X5X5X7 
3300=2X5X2X5X3X11. 
Any  number  less  than  1000  ending 
in  0  can  be  readily   factored  by  in- 
spection. 

3.  Factor  240,  340,  360,  420,  720, 
630,  640,  660,  490,  390. 

4.  Factor  800,  900,  1200,  1800, 
1600,  3100,  44,000,  75,000,  84,000, 
640,000. 

DIVISIBIIITY  OF  NUMBERS. 

There  are  certain  tests  by  means 
of  which  it  may  be  quickly  deter- 
mined whether  or  not  a  number  is 
divisible  by  certain  other  numbers, 
as  2,  5,  4,  etc.  These  tests  are  all 
based  on  the  fact  that  numbers  are 
written  on  the  scale  of  ten,  i.  e.,  that 
ten  ones  make  ten,  ten  tens  make  a 
hundred,  etc. 

1.  Two  will  divide  a  number  if  it 
ends  in  0,  2,  4,  6,  8. 

2.  Five  will  divide  a  number  if 
it  ends  in  0  or  5. 

3.  Ten  will  divide  a  number  if  it 
ends  in  0. 


48 


ARITHMETIC 


If  the  right  hand  digit  is  0  the 
number  is  made  up  of  tens.  370=37 
tens,  6430=643  tens.  Since  two, 
five,  and  ten  each  divides  ten  it  will 
divide  37  tens  or  370,  643  tens,  or 
6430.  Two  will  likewise  divide  370 
plus  6  or  376,  6430  plus  8,  or  6438. 
Five  will  divide  370  plus  5,  or  375, 
and   6430  plus   5,   or   6435. 

Applying  the  above  principles  fac- 
tor the  following  numbers : 

162,  98,  126,  132,  176,  175,  165, 
105,  1150,  1450. 

4.  Four  will  divide  a  number  if 
it  will  divide  the  number  expressed 
by   its   two   right   hand   digits. 

5.  Twenty-five  will  divide  a  num- 
ber if  it  ends  in  00,  25,  50  or  75. 
Principles  4  and  5  depend  upon  the 
facts  that  four  and  twenty-five  will 
each  divide  100,  and  that  if  the  num- 
ber ends  in  00  it  is  made  up  of  hun- 
dreds. 

Factor  the  following  numbers  by 
applying  principles  4  and  5 : 

168,  232,  216,  288,  324,  375,  675, 
825,  950,  625,  1125. 

In  factoring  always  use  the  form 
given  at  the  beginning  of  this  chap- 
ter. If  it  is  necessary  to  take  out  a 
factor  by  division,  do  it  by  using 
side  work. 

100^25=4.  Hence  200-^25=8, 
300-^25=12.  700-1-25=7X4=28 

1500-^-25=15X4=60. 

It  will  be  seen  from  these  exam- 
ples that  when  a  number  ending  in 
00  is  divided  by  25  the  quotient  may 
be  obtained  by  omitting  00  and  mul- 
tiplying by  4. 

Write  the  quotient  obtained  by  di- 
viding the  following  numbers  by  25 : 
17600,  1300,  15200,  19000,  26000, 
34700,  29600,  76500,  43900. 

1275=1200-f  75  ;  hence  1275-^25 
=48  f  3=51 .  2950=29004-50  ;  hence 
2950-f-25=4X  29-1-2=1 18.  14775 

-f-25=4X  147+3=591. 

Divide  by  25:  775,  925,  1350, 
5375,  6750,  4125,  43775,  54300, 
27625,  49350. 

6.  Nine  will   divide  a  number   if 


the  sum   of  its   digits   is   divisible   by 
9. 

7.  Three  will  divide  a  number  if 
the  sum  of  its  digits  is  divisible  by 
3. 

8.  Six  will  divide  an  even  num- 
ber if  the  sum  of  its  digits  is  divis- 
ible by  3. 

Applying  tests  6,  7  and  8,  deter- 
mine whether  3,  6  or  9  will  divide 
the  following  numbers :  474.  684, 
543,  856,  7485,  8694,  571,  735682, 
88875,  77772. 

Factor:  147,  693,  198,  234,  363, 
251,  2430,  294,  396,  16200. 

Whether  or  not  a  number  is  divis- 
ible by  7,  11,  or  13  or  any  larger 
prime  number  is  best  determined  by 
trial.  If  the  prime  numbers  are 
tried  in  the  order  of  their  size  it 
will  never  be  necessary  to  try  a 
prime  number  for  a  number  less  than 
the  square  of  that  prime  number.  If 
no  factor  is  found  the  number  is 
prime. 

Try  all  the  odd  numbers  not  end- 
ing in  5  between  50  and  100.  Write 
the  factors  of  the  composite  ones. 

Determine  the  primes  and  factor 
the  composites:  343,  199,  187,  171, 
299,    323,   209,   289,   371,   357. 

GREATEST  COMMON  DIVISOR. 

In  order  to  find  the  greatest  com- 
mon divisor  of  two  or  more  num- 
bers one  should  first  determine  the 
prime  factors  of  one  of  them.  These 
factors  may  then  be  tried  in  the 
other  numbers  and  those  which  are 
common  determined.  The  product 
of  the  prime  factors  that  are  com- 
mon will  be  the  greatest  common 
divisor. 

Find  the  greatest  common  divisor 
of  84.  105,  147,  189. 

84=^X^X3X7 

Therefore  3X7=21,  G.  C.  D. 
First  factor  84  as  shown  above. 
Then  try  these  factors  2.  3,  and  7 
in  the  numbers  105,  147  and  189. 
By  applying  the  tests  for  divisibility 
to  2  and  3,  trying  7.  it  is  found  that 
2   will   not   divide   each   of  the   num- 


LEAST  COMMON  MULTIPLE 


49 


bers  and  3  and  7  will.  Cross  out  the 
2's  and  underscore  3  and  7,  and 
the  product  of  3X7,  or  21,  is  the 
G.  C.  D. 

Find  the  greatest  common  divisoi 
of  the  following-  sets  of  numbers : 

1.  78,  130,  156,  208. 

2.  70,  175,  210,  315. 

8.     52,  91,  143,  221,  260. 

4.  95,  133,  247,  437. 

5.  99,  231,  451. 

Sometimes  the  same  factor  is 
common  two  or  more  times.  In  such 
case  divide  each  of  the  numbers  ex- 
cept the  one  factored  by  this  factor 
till  it  can  be  ascertained  how  many 
times  it  is  common. 

Find  the  greatest  common  divisor 
of  120,  168,  252,  432. 

120=2X^X2X2X3 


2)168- 


-252 432 


2)  84- 


-126 216 


42- 


63- 


-108 


First  factor  120  obtaining  2,  5,  2, 
2.  3  as  the  prime  factors.  Then  di- 
vide  each   of  the   other  numbers   by 

2  twice  in  succession  getting  the 
quotients  42,  63  and  108.  2  will  not 
divide  these  numbers  and  5  will  not, 

3  will.  Cross  out  2  and  5  and  un- 
derscore 3. 

2X2X3=12,  G.   C.  D. 

Find  the  greatest  common  divisor 
of:  (1)  81,^108,  135,  567.  (2)  98, 
112,  490,  336,  266.  (3)  75,  45,  195, 
435.  (4)  44,  220,  264,  198,  681. 
(5)    117,   195,   273,   351. 

Greatest  common  divisor  is  used 
in  reducing  fractions  to  their  lowest 
terms.  The  result  is  the  same 
whether  the  numerator  and  denomi- 
nator are  divided  at  once  by  their 
greatest  common  divisor  or  the  com- 
mon factors  are  removed  in  succes- 
sion. Reduce  ^^%09  to  its  lowest 
terms.  190=2X5X19.  2  and  5  are 
not  common.  Divide  by  19  and  the 
answer    ^%i    is    obtained.      ^^%o9  = 

Reduce  the  following  fractions  to 
their  lowest  terms: 


1. 

2. 
3. 
4. 
5. 

^%45. 
«%54. 

I«y399. 

^%24. 

•» 

Common  factors  may  be 

cast  out  in 

indicated  division. 

2^7 

;^x;^x3^ 

=7  X  2  X  2  =  28 

Ans. 

LEAST  COMMON  MULTIPLE. 

It  is  evident  that  a  multiple  of  a 
number  must  contain  all  its  prime 
factors,  and  that  a  common  multiple 
of  the  two  or  more  numbers  must 
contain  the  prime  factors  of  each 
and  contain  each  prime  factor  as 
many  times  as  it  occurs  in  any  one 
of  the  numbers. 

To  find  the  least  common  multiple 
of  24,  36,  60  and  72. 

24=2X2X2X3 
36=2X2X3X3 
60=2X5X2X3 
72=2X2X2X3X3 
2X2X2X3X3X5=360    L.  C.  M. 
The   work   of  multiplying  may   be 
shortened  by  taking  one  of  the  num- 
bers  and   multiplying   it  by  the   fac- 
tors of  the  other  numbers  not  found 
in   it.     Thus,   in  the  problem   solved 
above,  take  72  which  contains  2,  2, 
2,  3,  3,  and  multiply  it  by  5.     72X5 
=360  Ans. 

Find  the  L.  C.  M.  of  28,  42,  63. 

28=2X2X7 

42=2X3X7 

63=3X3X7 

28X3X3=28X9=252     Ans. 

Find  the  L.  C.  M. : 

1.  36,  54,  81. 

2.  16,  24,  72. 

3.  35,  42,  56. 

4.  38,  57,  54. 

5.  45,  60,  84,  126. 

Sometimes  the  factors  of  each 
number  may  be  seen  by  inspection. 
In  such  examples  it  is  unnecessary 
to  write  out  the  factors. 

Find  the  L.  C.  M.  of  15,  21,  35, 


50 


ARITHMETIC 


42.  It  may  be  readily  seen  that  the 
factors  2,  3,  5,  7  occur  not  more 
than  once  in  the  numbers.  The  L. 
C.  M.  is,  therefore,  42X5=210  Ans. 
Find  the  L.  C.  M.  of  the  following 
without  writing  the  factors  of  each 
if  possible. 

1.  18,  21,  24. 

2.  22,  33,  55. 

3.  35,  45,  55. 

4.  28,  24,  42. 

5.  6,     8,  10,  15,  25. 

If  one  of  the  given  numbers  is  a 

factor  of  another  it  may  be  omitted. 

Find  the  L.  C.  M.  of  14,  21,  35, 

42.     Omit  14  and  21   for   each  will 

divide  42. 

35=5X7 
42=2X3X7 
35X2X3=35X6=210 
Find  the  L.  C.  M. : 

1.  2,     3,    4,     6,     8,     9. 

2.  7,  14.  28,  42,  84. 

3.  15,  30,  45,  75. 

4.  8,  16,  24,  36,  48. 

5.  22,  44,  66,  88,  99. 

The  least  common  multiple  of 
small  numbers  may  usually  be  found 
by  inspection.  This  plan  should  be 
followed  when  possible. 

Find  the  L.  C  M.: 

1.  3,    4,     8,  12. 

2.  4,     8,  16,  20. 

3.  6,     9,  12,  18. 

4.  7,  14,  21. 

5.  15,  25,  30. 

Least  Common  Multiple  is  of  use 
in  reducing  fractions  to  a  common 
denominator  as  a  preparation  for 
addition  or  subtraction. 

Reduce  ^,  %2,  ^Vis,  %,  to  their 
least  common   denominator. 

The  L.  C.  M.  of  8,  9,  12,  18  is  72. 

%    =1%2 

%2  =  ^%2 

"/l8  =  ^y72 

Reduce  to  least  common  denomi- 
nator, finding  the  common  denomi- 
nator by  inspection  if  possible. 

2.       5/8,    746,    %,    Va,    1%4. 

3.  Yz,  Vs,  ri5,  %■ 

5.       3/5,    5/^,    %4,   7/^0,    2%5. 


ADDITION  AND  SUBTRACTION  OF 
FRACTIONS. 

A  good  form  is  helpful  in  addi- 
tion and  subtraction  of  fractions. 
The  one  here  given  is  compact,  bus- 
inesslike, and  free  from  objection- 
able features,  and  should  be  used 
exclusively  after  its  significance  is 
understood  through  the  use  of  the 
full  form. 

Add:   43/7+7/5+9K^+17%4+ll%o 
70 


30 
28 
35 
25 
49 


4% 

7  % 

9  1/2 
17  %4 
lL%o 

502%ol  i«%o 

167/,o==227/,o 

The  final  answer  50-%o,  is  the 
last  thing  written.  Any  other  work 
besides  that  here  written  should  be 
done  by  inspection  or  as  side  work. 

Add: 

1.  7^+93^+4f^-fll!:4o. 

2.  27H-f95^+47^-h5Ko. 

3.  %   +  40>^  -f  94/15  +  6243/10 
+15yi2. 

4.  26^+47r7+4i%8+ri4. 

5.  417%i+4%o+70%o+29^%. 
When  there  are  few  fractions  and 

the    common    denominator    is    small 
all  the  work  may  be  done  by  inspec- 
tion and  only  the   fractions   and  the 
result  written. 
Add  36,  433^,  18^,  and  6^. 
36 

43^ 

18^ 

6% 


105>^ 
Add: 

1.  32.>4+406/+82^-fl734. 

2.  8^+12>^-h260>^-f507. 

3.  433/   -f  18^  +  16%5 -f  200/s 

-f60. 

4.  863/^-f25M+624^+47%2. 

5.    26M-f6/5+7074o+429_/2+^. 
The    same    form    is    used    in    sub- 
tracting a  fraction. 


FRAC^'foS?  ^^ 


11 


51 


vSubtract  1754  from  43%. 


Subtract 


28 

433/4 

174/7 

21 
16 

26%  1 

%i 

1. 

2. 
3. 
4. 


2/ 


O. 


18213/8  _56%2. 
260%  — 184. 
2357/5-68%. 
902%4  — 3643/8. 
When    the    fractional   part    of    the 
subtrahend    is    larger    than    the    cor- 
responding  part   of   the   minuend,   a 
similar   method   is   used. 
Subtract  4%  from  83^. 
1     18 


8  ye     I       3 
4  %     I     10 


311/8  I  "/is 

One  is  taken  from  the  8  and 
changed  to  18th,  giving  ^%s,  the 
numerator  being  the  same  as  the 
number  above  3.  10  taken  from  18 
leaves  8  and  8  plus  3  equals  11. 

Subtract 

1.  72%  — 38^. 

2.  174^  —  897/2. 

3.  500  —  43y7. 

4.  294iy24  — 49^. 

5.  143/-^. 

Give  the  results  in  the  following 
by  inspection: 

1.  Ya  —  H- 

2.  UVe  —  ^V2. 

3.  26^  —  17^. 

4.  13^  —  65/2. 

5.  20%  — 8%. 

6.  48  —  37%. 

7.  27>^  — ^. 

8.  65%  —  57%. 

9.  733/1  —  641^. 
10.  18^  —  6742- 

To  Multiply  a  Fraction  by  a  Whole 
Number. 

In  multiplication  and  division  of 
fractions  it  is  best  to  memorize  a 
clear  concise  rule. 


To  multiply  a  fraction  by  a  whole 
number,  multiply  the  numerator  by 
the  whole  number  and  place  the  re- 
sult over  the  denominator. 

Multiply  %  by  5. 

5xr7  =  ^%=4%. 
Multiply : 

1.  y&  by  9.  4.     2%  by  14. 

2.  15/6  by  12.       5.     15/6    by    18. 

3.  1^  by  6.  6.     11/21  by  15. 
The    work    may   be   indicated   and 

the  process  often  shortened  by  can- 
cellation. 

Multiply  2^   by  28. 
7 

28  X  m  ^^^X19=i3%  =  66^. 

2 

Multiply : 

1.  %2  by  18.        4.     53/28  by  32. 

2.  yi4  by  35.         5.     1%   by  8. 

3.  27/9  by  15.        6.     17%  by  9. 
If   the    integral    part    of    a   mixed 

number  is  large  it  is  not  best  to  re- 
duce to  an  improper  fraction  before 
multiplying.  Where  the  multiplier 
consists  of  but  one  figure  the  reduc- 
tion and  carrying  can  be  done  at 
once. 

Multiply  27^  by  7.  27^ 

7 


194^ 

Multiply : 

1.     1265/  by  9. 

2.    237^    by    6. 

3.     465/2  by  7. 

4.     26494^1   by  8. 

5.     42857/2  by  5. 

If   the   multiplier 

consists   of  two 

or  more  digits  it  is 

best 

to  multiply 

the  fraction  and  whole 

number  sep- 

arately. 

Multiply  67^  by 

29. 

67^ 
29 

18^ 

603 

134 

1961>g 


52 


ARITHMETIC 


Multiply : 

1. 

2. 

3. 

4. 

5. 


by  28. 
757/12  by  32. 
2674%5  by  25. 
1960%!   by  24. 
36027i6  by  48. 


To  Divide  a  Fraction  by  a  Whole 
Number. 

To  divide  a  fraction  by  a  whole 
number  divide  the  numerator  or  mul- 
tiply the  denominator  by  the  whole 
number. 


Divide  4j4  by  6. 

4^-f-6  =  24/. 

Divide  54  by  3 


6  =  41 


If  each  fourth  is  divided  by  3  it  will 
be  seen  that  the  whole  is  divided 
into  twelve  parts.     Hence 

_  _M-^3  =  i/i2. 
Divide : 


2R 


%  by  4. 


3%  by  7. 


1 
2 

3.     8^7  by  12 
Divide  1%  bv  3. 

l%--3  =  i%--3 
Divide : 


4.  834  by  5. 

5.  12y9   by  7. 

6.  671/11   by  9. 


'V2, 


1.  %o   by  4. 

2.  2%  by  5 

3.  4%  by  6. 

4.  12^  by  8. 

5.  6>4  by  5. 

The  operation  may  be  indicated 
and  the  work  often  shortened  by 
cancellation. 


Divide  6^4  by  12. 
6^-f-12=27/^_._32=^ 


^ic- 


4x;^ 


Divide : 

1.  1^,5  by  21.       4.     7^   by  9. 

2.  7%  bv  10.        5.     44^^g  by  33. 

3.  16^3   by  20.     6.     71/11   by   39. 
When  the  integral  part  of  a  mixed 

number  is  larger  than  the  divisor  it 
is  not  best  to  reduce  to  an  improper 
fraction  at  first.  If  the  divisor  is 
small  use  short  division. 


Divide  187%  by  8. 
23% 


-8="%^8=  % 


8)187% 
Side   work 

3%- 
Divide : 

1.  43^  by  7. 

2.  29^  by  9. 

3.  265%  by  8. 

4.  1982^  by  8. 

5.  6371%  by  5. 

6.  7681^  by  6. 

If  the  divisor  is  large  use  long  di- 
vision. 

Divide  6744%  by  17. 
396122/53 


17)6745    % 
51 

164 
153 

115 
102 

13    % 
Side  work : 
13%  =  122/, 

122/,-..17--=122/,.3 

Divide : 

1.  864/    by    i4_ 

2.  2763^   bv  23. 

3.  1679^/,.,  by  18. 

4.  3462''i/o  by  26. 

5.  7632K  by  15. 

6.  87652 j/ii  by  21. 

To   Multiply   a  Whole   Number  by   a 
Fraction. 

Since  the  product  of  two  factors 
is  the  same  no  matter  which  is  used 
as  the  multiplier  the  rule  for  multi- 
plying a  fraction  by  a  whole  number 
may  be  used.  When  the  whole  num- 
ber is  large  it  is  frequently  desirable 
to  use  the  fraction  as  the  multiplier. 

To  multiply  a  whole  number  by  a 
fraction  multiply  the  number  by  the 
numerator  and  divide  the  product  by 
the  denominator: 

Multiply   47    by    ^v,. 
^X47=i4i/8=175/^     Ans. 


FRACTIONS 


53 


Multiply : 

1.  65  X%. 

2.  87  X%. 

3.  432  X%. 

4.  76X%i. 

5.  43X1%. 

6.  216  X  2%. 

It  is  not  necessary  and  frequently 
not  desirable  to  reduce  a  mixed  num- 
ber  to   an   improper    fraction   before 
multiplying.     It  is  best  in  such  cases 
to    multiply    by    the    fractional    part 
first. 
Multiply  89  by  27^. 
89 
27% 


Multiply : 


8)267 


Multiply : 


623 
178 

2436^ 


1. 
2. 
3. 
4. 
5. 
6. 


27  by 


%2. 


44  by  1%. 

45  by  %5. 
88  by  yg. 
496  by  29%.       . 
57851  by  43%. 

To  multiply  a  fraction  by  a  frac- 
tion multiply  the  numerators  for  the 
numerator  and  the  denominators  for 
the  denominator  of  the  product. 

Multiplying-  a  fraction  by  a  frac- 
tion is  best  explained  by  the  use  of 
a  diagram.     Multiply  %  by  ^. 


ri 


Yzoi  ys=  Yio,    hence 

y2xy5=yio^ 

Similarly  i^  X  ^  =  %o- 
In  like  manner  it  may  be  shown  that 

%x%  =  %5;  %x%  =  i%2;  etc. 

Have  the  pupils  find  the  products 
by  using  the  diagram.  By  compar- 
ing the  results  with  the  factors,  the 
rule  may  be  deduced. 

The  operation  should  be  indicated 
and  cancellation  used  when  possible. 
As  a  rule  mixed  numbers  should  be 
reduced  to  improper  fractions  before 
multiplying. 


1. 
2. 
3. 
4. 
5. 
6. 


%  by  %4. 
^%5  by  ;%8. 
279  by  3%o. 
%i  X  Ws^  X  1%, 


784  /\     726« 

2%  X  Wi  X  7%. 

4^11  X  5%  X  7%o  X  3%. 

A  compound  fraction  is  reduced  to 
a  simple  fraction  by  multiplication. 
Reduce  %  of  %2  of  6%  to  a  simple 
fraction : 

5      7      27 

%  of  %2  of  634=-X— X— 
^     l^      4 
4 
=  3%6  =  23/ie 
Simplify : 


1. 

% 

of 

4%. 

2. 

% 

of 

6%  +  %i  of 

2%4. 

3. 

H 

of 

2H 

Of   %. 

4. 

% 

of 
2% 

2% 

Of  iyi5 

-% 

of 

'" 

•^25 

of 

W2t 

B  +  %! 

of    6H 

— 

When  the  integral  part  of  a  mixed 
number  is  large,  it  is  best  not  to  re- 
dure  to  an  improper  fraction.  In 
such  cases  multiply  thru  by  the 
numerator  of  the  multiplier  and  di- 
vide the  result  by  the  denominator. 


Multiply  286%  by  %. 
286% 
% 


(Side  work) 

7%  =  5% 
^%^9=   % 


9)2293%     multiplying  by  8. 
254%     dividing  by  9, 


Multiply : 


1. 
2. 

3. 
4. 
5. 
6. 


796%  by  %. 
384%i  by  %. 
49%3  by  %i. 
3492%  by  %. 
4628%5  by  %. 
9371K4  by  %i. 


If  the  multiplier  is  a  mixed  num- 
ber, multiply  through  by  the  fraction 
as  shown  above  and  then  by  the  in- 
tegral part. 


54 


ARITHMETIC 


Multiply  679%  by  8%.     (Side  work) 
679  % 

46/7  =  34/^ 


8  % 


^^7-9  =  3% 


'63 


9)3397  %     1  Multiplying  by  5 


37734/63  I  34 
5436  %     I   36 


5814  1/9     I   '%s  =  l% 
If  the  integral  part  of  the  multi- 
plier   is    large    multiply    the    fraction 
by  it  and  write  the  result  as  a  part 
of  the  product,  then  muliply  the  in- 
tegral parts  together. 
Multiply  4758%  1  by  37%. 
4758  %i 
37% 


4)  14275i9i'i     Multiplying  by  3 

35684344     dividing   by   4 

23  %'i 
33306 
14274 


1796382%4 


(Side  work) 
3i%i  =  ^%t 

^%l  ^4  =  43/^4 

37X%i  =  25%i 
-236/11 

Multiply : 

1.  416%  by  7%. 

2.  36523/4  by  9^7. 

3.  6942%  by  83/3. 

4.  5873%!  by  251/. 

5.  32546/3  by  437/2. 

6.  27353/g  by  369%. 

To    Divide    a    Whole    Number    by    a 
Fraction. 

In  order  to  find  how  many 
times  4  ft.  is  contained  in  8  yards 
it  is  necessary  to  change  both  num- 
bers to  the  same  kind. 

8  yds.=  24  ft. 
24  ft.  ^4  ft.=    6      Ans. 
Divide  10  ft.  by  15  in.: 
10ft.  =  120  in. 
120  in.-f-15  in=      8       Ans. 
In    like    manner    when    a    whole 
number  is  to  be  divided  by  a   frac- 


tion the  whole  number  should  be  re- 
duced to  a  fraction  having  the  same 
denominator  as  the  divisor.  Hence 
the   rule : 

To  divide  a  whole  number  by  a 
fraction  multiply  the  whole  number 
by  the  denominator  and  divide  the 
result  by  the  numerator,  or  multiply 
the  whole  number  by  the  divisor  in- 
verted. 


Divide  28  by  %. 

,       28X9 


28 


■%  =  50H. 


Divide  56  by  51/: 

56  -f-  51/4  =  ^0X  ^  =  32/3  =10  2/3 

3 
Divide : 

1.  37  by  %. 

2.  42  by  2%. 

3.  78  by  73/. 

4.  5  by  7y2. 

5.  16  by  5^. 

6.  65  by   %   of  Gy^. 

If  the  whole  number  is  large  and 
the  denominator  of  the  fraction  is 
small,  write  the  divisor  to  the  left: 

Divide  4763  by  5^. 

Change  both  to  thirds : 

5%)  4763  840%7 


17    ) 14289 


17)14289 
136 


68 
68 


a 


Divide 


1.  2769  bv  2J4. 

2.  31483  bv  lYs- 

3.  7925  by  3%. 

4.  37261   by   IVc 

5.  17984  by  2%. 

6.  21358  by  7.>i^. 

In  the  exercises  in  division  of  frac- 
tions so  far  solved  the  entire  quo- 
tient including  both  fractional  and 
integral  parts  has  been  obtained.  It 
is  sometimes  desirable  to  have  only 
the  integral  part  of  the  quotient  and 
to  know  what  the  remainder  is.  The 
remainder  is  the  undivided  part  of 
the   dividend,    hence   is   of   the   same 


FRACTIONS 


55 


kind.  For  example,  if  dresses  each 
requiring  9  yards  are  made  from  a 
piece  of  cloth  containing  67  yards, 
7  dresses  can  be  made  and  there  will 
be  a  remainder  of  4  yards. 

The  remainder  may  be  found  in 
either  of  two  ways :  1st,  By  multi- 
plying the  divisor  by  the  integral 
quotient  and  subtracting  the  product 
from  the  dividend ;  2nd,  By  taking 
the  remainder  after  the  last  division, 
noting  what  it  is. 

In  the  last  model  given  above,  the 
integral  quotient  is  840  and  the  re- 
mainder is  %  or  3,  since  both  divi- 
dend and  divisor  were  changed  to 
thirds.  It  may  be  obtained  by  the 
first  way  given  above  as  follows : 
Dividend  4763 
840  X  5%  =  4760 


Give  the  entire  quotient  in  each 
of  the  examples  here  given,  also  the 
integral  quotient  and  remainder  in 
all  except  example  6. 

If  the  dividend  is  a  mixed  num- 
ber with  large  integral  parts  and  the 
comm^on  denominator  is  small,  write 
the  divisor  to  the  left: 

Divide  687 y2  by  22.5.  Reduce  both 
to  sixths : 

257i%6 


2%)  687 


Rem.  3 

Find  the  integral  quotients  and 
remainders  in  the  problems  in  last 
two  exercises  given  above. 


16     )4125 
32 

92 
80 

125 
112 

13 


To  Divide  a  Fraction  by  a  Fraction.         Divide: 


Ans.   25713^6,  or  257 
Quo.  and  2}i  Rem. 


If  the  fractions  have  a  common 
denominator,  divide  the  numerator  of 
the  dividend  by  the  numerator  of  the 
divisor. 

Divide  7^  by  ji- 
7y5-^V5  =  ^%-^ys  =  9    Ans. 
Divide : 

1.  2%5  by  3/35.     4.     %  by  %. 

2.  6^   by  H-       5.     %  by  4:%. 

3.  18^  by  3^.     6.     7%  by  21/9. 
Give    the    entire    quotient    in    eacte 

example  here  given ;  also  the  inte- 
gral quotient  and  the  remainder  in 
all   except   example   5. 

If  the  common  denominator  is 
small,  reduce  the  fractions  to  a  com- 
mon denominator  and  divide  as 
above. 

Divide  7^^  by  1%. 

V/2-^l%  =  ^%^%  =  Q     Ans.- 

Divide : 

1.  8^  by  2%. 

2.  15^  by  7/12. 

3.  14^5  by  8/15. 

4.  ii/s  by   54- 

5.  Yf)  bv  yi. 

6.  6^  by  8,^. 


1.  2647%  by  1^. 

2.  675845'^    by   3><. 

3.  473^   by  7y2. 

4.  358^   by  2^. 

5.  627%  by  4>^. 

6.  3721^  by  7^. 

A  fraction  may  be  divided  by  a 
fraction  by  applying  the  following 
rule : 

To  divide  a  fraction  by  a  fraction, 
invert  the  divisor  and  proceed  as  in 
multiplication. 

Divide  42/^  by  %: 

4%  --  %  =  ^%  X  %  =^%  =  e%. 

When  problems  are  solved  in  this 
manner,  cancellation  may  be  used 
and  the  entire  quotient  only  is  ob- 
tained. If  the  integral  quotient  and 
the  remainder  are  desired  the  re- 
mainder may  be  found  by  multiply- 
ing and  subtracting  as  previously 
given. 

I  28 

Dividend      4  %  |     8 

6X%  =  S%  1  21 


l%8l  1%8 


56 


ARITHMETIC 


Hence    the    integral   quotient    is    6 
and  the  remainder  ^%8- 
Divide : 

1.  %  by  %. 

2.  S%  by  5344. 

3.  %  of  6%  by  71/2. 

4.  5/8  of  12%  by  H  of  25/14. 

5.  7y3  by  %  of  63/^. 

6.  ^  of  %  of  33^^  by  •%  of  4J^. 

A  complex  fraction  is  merely  an 
indicated  divisor  in  which  the  num- 
erator is  the  dividend  and  the  de- 
nominator the  divisor.  It  can  be 
simplified  by  performing  the  indi- 
cated operations. 

Q.      yf    63/7 
Simplify  -^^ 

073 

9 


Simply 
1      - 


5 


Vs 


2. 


3^ 

%  of  42/5 

V/2+m 


5^ 


4^3 


12^  +  4^ 

%  of  syg-iKo 

14-2/3    of   5y4 


%  X  1% 


THE   USE    OF   SIGNS. 

1.  If  only  the  signs  -|~  ^^'^  — 
occur  in  an  expression  the  opera- 
tions are  to  be  performed  in  order 
from   left  to   right. 

6  4-  7  —  3  —  2  -f  4  =  12. 

2.  If  only  the  signs  X  and  -f- 
occur  in  an  expression  the  opera- 
tions are  to  be  performed  in  order 
from  left  to  right. 


48  ^  6  X  2  =  16 
48  -f-  2  X  6  =  144 
48  -f-  6  -^  2  =      4 

3.  If  the  signs  -}->  — >  X,  ana 
-f-  occur  in  an  expression  the  mul- 
tiplications and  divisions  are  to  be 
performed  first,  and  then  the  addi- 
tions  and   subtractions. 

12+3  X  2—9^3=12+6—3=15 
Find  the  value: 

1.  4X7—8^-2+6X3. 

2.  27—12^3+2. 

3.  18+9X4^6—4. 

4.  16X3+24-=-8X2— 18-^6. 

5.  7X5+10—4X5+3. 
Simplify : 

,       ^VaXIVi+V^  of  2>4-i%8X2. 


13/28  of  2+yi  of  214-IM  of  1% 

Ans.  20. 


2.    2Mx^g^-f^-x     ^^Z- 


6%6+7  %        1^5X91/11 

Ans.  %5 
3. 

Ye  of  Ii3/i6+lj^  of  Q}i-iy3  of  5% 
Ye  of  2^  of  5^ 

Ans.  Yi 

4.    ^X^^+gj^-^^o         Ans.r8 


5y9-7^8H-28%o+^ 
69/4+5^  X3y7—7^ 
3K5+2>^-4yio 


Ans.  1055/112 


6  ^^-^^ y2z   of 

^'    1/5  of  5/,-^10/3  ^^'   °* 
1/2  of  4y9 


13^^  of  5y3 

15^—^  of  1:^  of  m 


Ans.  423/^4 
Ans.  125/47 


iy20— %  of  2^ 

2MX%8  of  3/5X1/4X1/5. 

Ans.  1>^ 

o    ^x^^2+^— 3/o^y2oX^ 


—  X~-f-20 
^3      H 

Simplify  the  following: 


Ans.  1 


1.       1%    of    l0/i7 


5^— 3>4 


4^+25/ 
+%o  of  3^.     Ans.  4yio 


DECIMALS 


57 


2. 

3. 

4. 


i%o+i%3+3ri6xiyi9 

3%--2%-4%^10^ 
3K— 2^ 


Ans.  6%o 


434— 3>^ 


+%    of    3.)/5-iy20--l%5 


.214o  Alls.  %5 

4yi2Xiiy4o-3%4^5y7 

(7%+6^8)^(8yi2-5%8) 

Ans.  14%40 

22/.  V 123/- 22/_\/27/  . 


^  %5  of  18%+85/^-4K6 

4">^~3i%6-^i4y4o+iy9 


Ans.  1^^ 


Vs  of  22/2    ^  82^-^-^        , 


5^X27/12 


'3        69H^3/i6 

Ans.  5%.,2 


+ 


1% 


12 


■    8%X1/      3/5   of  1% 

— 2yi2--2iyi7    Ans.l/ 

g     ^y63-f-^%69+5iyi5-^iy27  ,  „„    17/ 

6iyi8X33/2o-^iyi6X22/9.'^"''-'/i23 

DECIMALS. 

In  decimal  fractions  the  denomina- 
tor is  10,  100,  1000,  or,  in  general, 
1  with  one  or  more  ciphers  annexed. 
It  is  not  written,  but  is  indicated  by 
a  dot  called  a  decimal  point  placed 
in  the  numerator. 

The  number  of  places  to  the  right 
of  the  decimal  point  in  the  numera- 
tor is  the  same  as  the  number  of 
ciphers  after  the  one  in  the  denomi- 
nator. Thus,  in  .4,  .37,  6.145  the  de- 
nominators are  10,  100  and  1000, 
respectively. 

If  the  decimal  has  no  other  frac- 
tion attached  it  is  a  pure  or  a  mixed 
decimal. 

In  a  pure  decimal  the  numerator 
is  less  than  the  denominator.  Hence 
there  are  no  significant  figures  to  the 
left  of  the  decimal  point.  .17,  .809 
and  .0653  are  pure  decimals.  These 
correspond  to  proper  fractions. 

In  a  mixed  decimal  the  numerator 
equals  or  exceeds  the  denominator. 
Hence  there  must  be  some  significant 
figure  or  figures  to  the  left  of  the 
decimal  point.     3.7,  60.08,  1.2374  are 


mixed  decimals.  They  correspond  to 
mixed  numbers  and  improper  frac- 
tions. 

In  reading  pure  decimals,  read  the 
numerator  and  then  give  the  name 
of  the  denominator.  Thus,  .26  is  read 
twenty-six  hundredths,  .0026  is  read 
twenty-six  ten-thousands,  .00326  is 
read  three  hundred  twenty-six  hun- 
dred-thousands. 

Read:  .704,  .0470,  .600,  .00006, 
.0400,  .410,  .000910,  .0000900,  .0572, 
.5072. 

Mixed  decimals  may  be  read  in 
two  ways :  as  a  mixed  number,  and 
as  an  improper  fraction.  When  read 
as  a  mixed  number  and  is  used  at 
the  decimal  point  only,  for  this  sep- 
arates the  integral  from  the  frac- 
tional part.  Thus,  4.9,  27.432,  600.- 
028  are  read  four  and  nine-tenths, 
twenty-seven  and  four  hundred  thir- 
ty-two thousandths,  six  hundred  and 
twenty-eight  thousandths,  respective- 
ly. In  reading  a  mixed  decimal 
as  an  improper  fraction,  read 
the  entir^e  numerator  as  it  would 
be  read  if  there  were  no  decimal 
point  and  then  give  the  name 
of  the  denominator.  Thus,  6.7, 
43.007,  75.0800,  9.60  are  read  67 
tenths,  43007  thousands,  750800  ten- 
thousandths,  and  960  hundredths,  re- 
spectively. 

Read  the  following,  first  as  mixed 
numbers  and  then  as  improper  frac- 
tions : 


60.009 

9.0009 

307.0705 

150.7209 

9.43075 

61.4 

600.7 

100.008 

600.007 

728.46 

72800.46 

15.09 

1500.09 

700.008 

5.9 

Write  the  following: 

1.  4     hundred-thousandths,      400 
thousandths. 

2.  3000  ten-millionths,   3010  mil- 
lionths. 

3.  805  tenths,  800  and  5  tenths. 


58 


ARITHMETIC 


4.  3076  hundredths,  3000  and  76 
hundredths. 

5.  500  ten-thousandths,  510  thou- 
sandths, and  500  and  10  thousandths. 

Annexing  a  cipher  to  a  decimal 
multiplies  both  numerator  and  de- 
nominator by  ten  and  therefore  does 
not  change  its  value.  Thus,  4.6= 
4.60=4.600=4.6000,  etc.  A  whole 
number  may  be  written  as  a  decimal 
by  placing  a  decimal  point  to  its 
right  and  annexing  one  or  more 
ciphers.  Thus,  43  =  43.0  =  43.00 
=43.000=43.0000,  etc.  Read  both 
as  a  mixed  decimal  and  as  an  im- 
proper fraction.  Reduce  21  to  tenths, 
to  thousandths,  to  milHonths. 

Dropping  a  cipher  at  the  right  of 
a  decimal  divides  both  numerator 
and  denominator  by  ten,  hence  does 
not  change  its  value.  Thus,  2.4000= 
2.400=2.40=2.4.     Read  each. 

Moving  the  decimal  point  one 
place  to  the  right  divides  the  de- 
nominator by  ten  without  changing 
the  numerator,  hence  multiplies  the 
fraction  by  10.  Name  the  numera- 
tor and  the  denominator  in  each  of 
the  following:  .000726,  .00726, 
.0726,  .726,  7.26,  72.6,  726.  Read 
each. 

Moving  the  decimal  point  one 
place  to  the  left  multiplies  the  de- 
nominator by  10  without  changing 
the  numerator,  hence  it  divides  the 
fraction  by  10.  Name  the  numerator 
and  the  denominator  in  each  of  the 
following:  38,  3.8,  .38,  .038,  .0038, 
.00038,    Read  each. 

Reduce  a  Decimal  to  a  Common 
Fraction. 

A  pure  decimal  may  be  changed  to 
a  common  fraction  by  writing  the 
denominator  under  the  numerator, 
omitting  the  decimal  point.  The 
fraction  may  then  be  reduced  to  its 
lowest  terms.     Thus,    .072="^ 7^  000= 

ri2.5. 

Reduce  to  common  fractions  and 
to  their  lowest  terms : 

.45  .144  .625 

.045         .00144         .6205 
.405         .10044         .00625 
Mixed    decimals    may     be     treated 


either  as  mixed  numbers  or  as  im- 
proper fractions.  Thus,  16.25= 
1625/100  =  16M  or 

16.25=1625/^^^:^65^=^16/4. 

Reduce  the  following  both  ways: 
6.56  43.75  341.8 

9.006  12.15  3.418 

90.06  1.215  2.760 

ADDITION    AND    SUBTRACTION    OF 
DECIMALS. 

Read  the  following:  .3,  .03,  .003, 
.0003.  It  will  be  seen  from  the 
above  that  a  figure  in  the  first  place 
to  the  right  of  the  decimal  point  rep- 
resents tenths ;  in  the  second,  hun- 
dredths ;  in  the  third,  thousandths ; 
and  so  on.  For  this  reason  the  or- 
ders to  the  right  of  the  decimal  point 
are  called  tenths,  hundredths,  thou- 
sandths, ten-thousandths,  etc.  Name 
the  order  represented  by  each  figure 
in  the  following:  4063.259187.  Note 
that  in  reading  a  decimal  the  name 
of  the  right  hand  order  is  given. 

Note  that  1=1.0,  .1=.10,  .01= 
.010,  .001=.0010,  and  in  general  that 
ten  units  of  any  order  makes  one 
of  the  next  higher.  Hence  in  addi- 
tion and  subtraction  of  decimals, 
carrying  is  done  the  same  as  in 
whole  numbers.  Care  must  be  tak- 
en that  units  of  the  same  order  stand 
under  each  other  and  are  added. 

Add: 


1. 

.247 

.6053 

.0958 

.86 

.09016 

2. 

86.094 

32.7005 

603.00951 

720.900084 

.78 

97. 

3.  8200 

-f  2.834  + 

,6005 

306.942. 

Subtract ; 

4. 

.784 
—  .329 

DECIMALS 


59 


5.     64.3 
—28.56 


970. 
—7.083 


7.  23.6—2.36. 

8.  7.4— .0659 

9.  .7— .0849. 

10.  543.27—57.904. 

mJLTIPLICATION  AND  DIVISION  OF 
DECIMALS. 

The  principles  that  apply  in  com- 
mon fractions  apply  also  in  decimals, 
and  the  same  rules  for  multiplication 
and  division  might  be  used.  It  is 
better,  however,  to  use  special  rules 
in  handling'  decimals. 

To    Multiply   a   Decimal   by   a  Whole 
Number. 

Irr  common  fractions  the  numera- 
tor is  multiplied  and  the  denominator 
is  not  changed.  Likewise  in  deci- 
mals the  number  written,  the  nu- 
merator, is  multiplied  by  the  whole 
number  and  the  same  number  of 
places  pointed  off. 

Multiply  29.67  by  34. 
29.67 
34 


11868 
8901 


1008, 
Multiply : 


1. 
2. 
3. 
4. 
5. 


.084  by  149. 
63.828  by  365. 
920.05  by  76. 
.475  by  357. 
.625  bv  96. 


To  Multiply  a  Decimal  by  One  with 
one  or  more  ciphers  annexed. 

Multiply  .4768  by  1000. 
.4768 
1000 


BULE — To  multiply  a  decimal  by 
one  with  ciphers  annexed,  move  the 
decimal  point  as  many  places  to  the 
right  as  there  axe  ciphers  annexed  to 
one. 

Note  that  this  is  merely  a  kind  of 
cancellation.  Write  the  result  at 
once.     Thus,  4.672X100=467.2. 

Multiply : 

1.  67.853  by  100. 

2.  6.3009  by  10<30. 

3.  7.48  by  10000. 

4.  .76  by  100. 

5.  .706  by  100000. 

To  Multiply  a  Decimal  by  Any  Num- 
ber with  Ciphers  Annexed. 

Multiply  .0674  by  4700. 
.0674 
4700 


471800 
2696 


316.7800  or  316.78. 

RTTLE:  Multiply  as  in  whole  num- 
bers and  move  the  decimal  point  as 
many  places  to  the  right  as  there  are 
ciphei^  annexed  in  the  multiplier. 

When  the  multiplier  is  not   larger 
than  12  with  ciphers  annexed,  write 
the  result  at  once.     Thus,  4.7954  X 
7000=33567.8. 
Multiply : 

1.     30.795  by  600. 
.07532   by   9000. 
48.73  by  80000. 
2.876  by  4300. 
672.9  by  128000. 
.00924  by  86000. 


2. 
3. 
4. 
5. 
6. 


476.800     or       476.8 


To  Divide  a  Decimal  by  a  Whole 

Number. 

One  way  of  dividing  a  fraction  by 
a  whole  number  is  to  divide  the  nu- 
merator by  the  whole  number  with- 
out changing  the  denominator.  Ap- 
plying this  to  decimals  would  give 
the  rule  here  stated. 


60 


ARITHMETIC 


RULE:  To  divide  a  decimal  by  a 
whole  number  divide  as  in  whole  num- 
bers and  point  off  as  many  decimal 
places  in  the  quotient  as  there  are 
decimal  places  that  have  been  used  in 
the  dividend. 

The  decimal  point  should  be 
placed  in  the  quotient  as  soon  as  it 
has  been  reached  in  the  dividend.  If 
the  quotient  is  placed  in  its  proper 
position  over  the  dividend,  the 
decimal  point  of  the  quotient  will 
come  directly  over  the  decimal  point 
of  the  dividend. 

Note  that  there  must  be  a  figure 
in  the  quotient  for  every  figure 
which  has  been  used  to  the  right  of 
the  decimal  point  in  the  dividend. 

Divide  2937.97  by  47. 
62.51  Ans. 


47)2937.97 

282 


117 
94 


239 
235 


47 

47 

Divide  4.6292  by  284. 
.0163  Ans. 


284)4.6292 
2  84 


1  789 
1  704 


852i 
852' 

Divide : 

1.  95.88  by  94. 

2.  3130.48  by  872. 

3.  .0201474  by  54. 

4.  19.7635  by  841. 

5.  3123.6  by  685. 

6.  332.45  by  488. 


The  divisor  is  not  always  exactly 
contained  in  the  dividend.  There 
may  be  a  remainder,  however  far  the 
division  is  carried.  In  such  cases 
one  of  three  things  may  be  done : 
The  division  may  be  carried  as  far 
as  desired  and  the  remainder 
dropped,  the  quotient  may  be  ob- 
tained as  far  as  desired  and  the  re- 
mainder noted,  or  the  remainder  and 
divisor  may  make  a  fraction  to  be 
attached   to  the   last  quotient   figure. 

Divide  185.72  by  345,  obtaining 
the  quotient  to  three  decimal  places. 
.538 


345)185.720 
172  5 

13  22 

10  35 

2  870 

2  760 

110 

"%45=2%9 

1st  Quotient    .538+. 

2nd  Quotient    .538  Rem. 

.110. 

3rd  Quotient   .5382%9. 

Expressions  like  the  third 
are  called  complex  decimals. 

quotient 

Divide    carrying    the    quotient    to 
two   decimal   places   and   express   the 
result  in  three  ways  as  shown  above: 

1.     45.7  by  21. 

2.     7695.83  by  492. 

3.     925  by  364. 

4.     904.68  by  283. 

5.     3.59  by  87. 

6.     3658.74  by  364. 

To  Divide  a  Decimal  by  one  witfc 
one  or  more  Ciphers  annexed. 

One  way  of  dividing  a  fraction  by 
a  whole  number  is  to  multiply  the 
denominator  by  the  whole  number, 
leaving  the  numerator  unchanged. 
This  may  be  applied  in  dividing  a 
decimal  by  one  with  ciphers  annexed 
as  follows: 


DECIMALS 


61 


RULE:  To  divide  a  decimal  by 
one  with  one  or  more  ciphers  an- 
nexed move  the  decimal  point  as  many 
places  to  the  left  as  there  are  ciphers 
annexed  to  the  one. 

Divide  267.43  by  100.  Answer 
2.6743. 

Divide : 

1.  639.4  by  10. 

2.  47.085  by  1000. 

3.  .0038  by  100. 

4.  286  by  100. 

5.  76.32  by  1000. 

6.  4.694  by   1000000. 

To  Divide  a  Decimal  by  any  num- 
ber with  ciphers  annexed,  cross  off 
the  ciphers  at  the  right  of  the  di- 
visor and  move  the  decimal  point 
in  the  dividend  as  many  places  to 
the  left,  then  divide  by  the  remain- 
ing figures  in  the  divisor  as  hereto- 
fore. 

It  is  best  not  to  move  the  decimal 
point,  but  to  indicate  by  a  mark  the 
place  to  which  it  should  be  moved. 
Do  not  fail  to  place  the  mark  before 
beginning  the  division  and  to  place 
the  decimal  point  in  the  quotient  as 
soon  as  the  mark  in  the  dividend  is 
reached. 

Divide  2973.6  by  1800. 
1.65  2 


18pp)29'73.6 
18 

11  7 
10  8 

93 
90 

3  6 
3  6 

Divide : 

1. 

679.35   by   15000. 

2. 

7.84  by  39200. 

3. 

.0018  by  4500. 

4. 

5.736  by  16000. 

5. 

57548.4  by  5460. 

6. 

72.9  by  270000. 

To  Reduce  a  Common  Fraction  to  a 
Decimal. 

A  fraction  is  an  indicated  division 
in  which  the  denominator  is  the  di- 
visor. Hence  a  fraction  may  be  re- 
duced to  a  decimal  by  annexing 
ciphers  to  the  numerator  and  divid- 
ing by  the  denominator.  A  decimal 
point  should  be  placed  after  the  nu- 
merator before  the  ciphers  are  an- 
nexed. 


educe  '^g  to  a  decimal: 

.4375     Ans. 

16)7.0000 
6  4 

60 

48 

120 

112 

80 

80 

educe  to  decimals: 

1.  Vs.                7. 

2.  %5.               8. 

3.  %o.               9. 

4.  ^%5o-          10. 

5.  11/32.              11. 

6.  i%2.5.          12. 

2%. 

"/2OO. 

^%oo. 

Most  fractions  will  not  make  ex- 
act decimals.  In  reducing  such  frac- 
tions to  decimals  the  result  may  be 
obtained  to  the  desired  number  of 
decimal  places  and  the  remainder 
dropped  or  the  result  may  be  writ- 
ten as  a  complex  decimal. 

Reduce  the  following  fractions  to 
decimals  of  three  decimal  places  and 
drop  the  remainder : 


1.  %. 

6.       %75. 

2.     Vis. 

7.    38y,,. 

3.     %T. 

8.     %5. 

4.     %3 

9.       -%.5 

5.      "/^9. 

10.       %87. 

Reduce   the 

following  to   complex 

decimals  of  four  decimal  places: 

1.  yi7. 

6.     ^1000- 

2.  ^%^. 

7.       '«%.3. 

3.  «y4i. 

8.       3000/^. 

4.       %7- 

9.     %i."' 

5.     %oi,. 

10.       ^%3. 

62 


ARITHMETIC 


A  complex  decimal  may  be  ex- 
tended to  any  required  number  of 
decimal  places  by  annexing  ciphers 
to  the  numerator  of  the  common 
fraction  and  dividing  by  its  denomi- 
nator and  annexing  the  result  to  the 
original  decimal  part. 

Reduce  the  6.7%3  to  a  complex 
decimal   of  three   decimal   places. 

Two  additional  places  are  required 
so  two  ciphers  are  annexed  to  the  5. 

.38%3 


13)5 

.00 

3 

9 

1 

10 

1 

04 

Hence    6 . 7%3  =-  6 .  7386/i3. 

Reduce  the  following  to  complex 
decimals   of  four  decimal  places : 

1.  .63%i.  6.  .143/70. 

2.  7.8%7.  7.  9.020/2.3. 

3.  673/1,.  8.  48%i. 

4.  .OOysi.  9.  .05/300. 

5.  .760%4.  10.  60.273/4,. 

In  complex  decimals  the  common 
fraction  belongs  to  the  order  to 
which  it  is  annexed.  Thus,  .26% 
means  .26  and  %  hundredths.  .0% 
is  read  %  tenths.  A  fraction  does 
not  represent  an  order  and  should 
never  be  immediately  preceded  by  a 
decimal  point. 

Write  %  hundredths;  4%  thou- 
sandths ;  %  tenths. 

A  complex  decimal  may  be  re- 
duced to  a  common  fraction  by  re- 
ducing the  numerator  to  an  improp- 
er fraction  and  writing  for  the  de- 
nominator the  denominator  of  the 
common  fraction  followed  by  as 
many  ciphers  as  there  are  decimal 
places. 

Reduce  .47%  to  a  common  frac- 
tion. 

473/7  =  332/^^  hence 


Reduce  to   common  fractions : 

1.  .63^.  6.     3.0%i. 

2.  .055%.  7.     .304/13. 

3.  .63%i.  8.     .62%! 

4.  8.07%.  9.     .4-/30. 

5.  .004/  10.     .163/. 

The  number  of  decimal  places  in 
a  complex  decimal  may  be  reduced 
by  treating  the  part  which  follows 
the  desired  stopping  place  as  a  com- 
plex decimal. 

Reduce  .0286%  to  a  complex  deci- 
mal of  two  decimal  places. 

It  is  required  to  reduce  the  .0086% 
to  a   fraction  of  hundredths, 

.86%  =  260/3^^  _i%g,  hence 

.0286%  =.0213/5. 

Note  that    .0086  is    .86%  of  a  hun- 
dredth. 

Reduce  to  complex  decimals  of 
one   decimal  place : 

1.  6.73%.  6.     7.6002%6. 

2.  .4363/.  7.     4.581%i. 

3.  .073/1.  8.     .000%3. 

4.  .80%.  9.     .793020/-. 

5.  .340%  10.  8.473/1. 
The  student  should  be  able  to  de- 
termine by  inspection  whether  or 
not  a  fraction  whose  denominator  is 
less  than  one  hundred,  or  even  one 
thousand  will  make  an  exact  deci- 
mal. 

A  common  fraction  is  reduced  to 
a  decimal  by  division.  The  numera- 
tor with  ciphers  annexed  must  ex- 
actly contain  the  denominator,  or, 
what  is  the  same  thing,  every  factor 
of  the  denominator  must  cancel  simi- 
lar factors  in  the  numerator  or  the 
quotient  will  be  fractional.  Annex- 
ing a  cipher  to  the  numerator  in- 
troduces the  prime  factors  2  and  5 
only,  and  additional  ciphers  merely 
increase  the  number  of  2's  and  5's. 

Determine  whether  or  not  %«, 
%8.  and  %2r,  will  make  exact  deci- 
mals. 

3X3X^X5X2X5X2X5X^X5 
9.0000 


16 


16 

2X^X2X^ 


5625. 


DECIMALS 


G3 


%8  = 


3X3X2X5X^X5X2X5 
9.000 


28    - 
2X2X7 
The  factor  7  cannot  be  introduced 
by   annexing  ciphers,   hence  %8   will 
not   make   an   exact   pure   decimal. 

3X3X2X/5X2X^X2XP 
9.000 
125 

Determine  by  inspection  which  of 
the  following  fractions  will  make 
exact  pure  decimals,  and  test  your 
conclusions : 

1.  yi8.  7.      27/.^. 

2.  "/24.  8.       33/^4. 


%25  = 


:.072 


15. 


%-■ 


4.     i%5. 


9-     Whs- 

10.       2%^. 
6.       2548.  12.      %25. 

If  the  fraction  is  in  its  lowest  terms, 
the  denominator  must  contain  no  prime 
factor  except  2"s  or  5's,  or  2's  and  5's 
in  order  to  reduce  to  a  pure  deci- 
mal. The  number  of  decimal  places 
is  the  same  as  the  number  of  2's  or 
5*s  in  the  denominator.  The  pow- 
ers of  2  under  1000  are  2,  4,  8,  16. 
32,  64,  128,  256.  512;  those  of  5  are 
5,  25.  125,  and  625.  These  num- 
bers, or  these  with  one  or  more 
ciphers  annexed,  are  the  only  num- 
h>ers  under  1000  which  being  used 
as  denominators  of  fractions  will 
make  pure  decimals  if  the  fractions 
are  in  their  lowest  terms. 

Write  all  the  denomhiators  under 
1000  of  fractions  that  will  make 
pure  decimals  and  tell  how  many 
decimal  places  will  be  required  foi 
each.     Test  your  answers. 

When  complex  decimals  are  to  be 
added  or  subtracted  they  must  first 
be  reduced  to  the  same  order. 

Add  47%  and  .27^7  and  .0954%i 
and  8.643^^4. 

I  21 


47  -;;   ==47.8333% 

.27     -H   =      .2757V7 


8 


.09541^M=      .0954yoi  | 
.643  %4=    8.64.36%     ! 


Add: 

1.  .45%i  -h  83^  +    .9731/22   -f 

7.324. 

2.  *%8   +    -lYii   +    .6281^    4- 

6.034/15. 

Add  and  subtract  as  the  signs  in- 
dicate : 

1.  43.73/7— .4373/7. 

2.  29 . 6%  -f  .  296%  —  296%. 

3.  8742/75  —  87.42/75—       .8742/75 
-8.742/75. 

4.  49 .  83/80  —  .  4983/80  -f  4983/80 


5.     23%-f  2.352/3 +  23.7% — 
.2375%. 

To  Multiply  a  Decimal  by  a 
Decimal. 

In  multiplying  common  fractions 
the  numerators  are  multiplied  to- 
gether for  the  numerator  and  the 
denominators  for  the  denominator  o 
the  product. 

In  a  decimal  the  denominator  is 
one  with  a  cipher  or  ciphers  an- 
nexed. To  multiply  two  such  num- 
bers together  we  merely  annex  the 
ciphers  of  one  number  to  the  other. 
Thus  100X1000=100000.  In  mul- 
tiplying two  decimals  together  then 
the  number  of  decimal  places  in  the 
product  equals  the  sum  of  the  deci- 
mal places  in  the  two  factors. 

To  multiply  a  decimal  by  a  deci- 
mal multiply  as  in  whole  numbers  and 
point  off  as  many  decimals  in  the 
product  as  there  are  decimal  places  in 
both  multiplicand  and  multiplier. 

Multiply   .4635  by   .093. 
.4635 
.093 


56.84812/21  I  2%i 


13905 
41715 

.04.31055 


64 


ARITHMETIC 


Multiply : 


1. 
2. 
3. 
4. 
5. 
6. 


.784  by  .932. 
.2653  by  .317. 
.0695  by  .01082. 
97.65  by  .6438. 
.03836  by  .837. 
.0907  by  470. 
Complex    decimals    are    multiplied 
in  similar  manner. 
Multiply  67.954%  by 
67.954% 
.09 


.09. 


6.11592% 
Multiply  7.65  by  .073% 
7.65 
.073% 


6)3825 


6371/2 
2295 
5355 


.56482K   or   .564825 
Multiply  6.743%  by  .28%. 


.6743  % 

.28  % 

7)     26974  %  1  21 

38531%!  1  11 

18  %  1  14 

53944        1 

13486          1 

.192676  y2iP%i 

Multiply : 

1. 

86.357%  by  .08. 

2. 

.0362%!  by  4.5. 

3. 

6.947  by  .004%. 

4. 

463.85  by  18. 5% j. 

5. 

1.058^  by  .07%. 

6. 

2.53%o  by  1.6^/^. 

To  Divide  a  Decimal  by  a  Decimal. 

If  the  divisor  and  dividend  have 
the  same  number  of  decimal  places 
they  will  have  a  common  denomina- 
tor, and  the  quotient  be  a  whole 
number  the  same  as  in  division  of 
common   fractions. 


Divide  4.688  by  .293. 
16 


,293)4.688 
2  93 


1  758 
1  758 


If  the  dividend  has  more  decimal 
places  than  the  divisor,  a  mark 
should  be  placed  in  the  dividend  cut- 
ting oft  as  many  decimal  places  as 
the  divisor  contains.  This  will  de- 
termine he  place  where  the  intergral 
part  of  the  quotient  ends  and  the 
decimal  part  begins.  The  mark 
should  be  placed  in  the  dividend  be- 
fore the  division  is  begun. 

Divide  4.7875  bv   .125. 
38.3 


125)4.787' 5 
3  75 


1  037 
1  000 


375 
375 

Care  should  be  taken  that  the 
quotient  is  placed  in  its  proper  posi- 
tion above  the  dividend  and  that  the 
decimal  point  is  placed  in  the  quo- 
tient as  soon  as  the  mark  in  the  divi- 
dend is  reached.  The  decimal  point 
in  the  quotient  will  come  directly 
over  the  mark  in  the  dividend  if 
these  precautions  are  observed. 
There  should  be  a  figure  in  the  quo- 
tient over  each  figure  used  in  the 
dividend  to  the  right  of  the  mark. 

If  the  dividend  contains  fewer 
decimal  places  than  the  divisor  has 
ciphers  should  be  annexed  to  the 
dividend  before  beginning  the  divi- 
sion. 

Divide  1137.6  by   .237. 
4  800 


,237)1137.600' 
948 


189  6 
189  6 


DECIMALS 


65 


These  directions  are  summed  up  m 
the  rule:  To  divide  by  a  decimal, 
Mark  off  as  many  decimal  places  in 
the  dividend  as  there  are  decimal 
places  in  the  divisor,  beginning  at 
the  decimal  point.  Divide  as  in  whole 
numbers,  placing  each  figure  of  the 
quotient  directly  over  the  right  hand 
figure  of  the  dividend  used  in  obtain- 
ing it.  Place  the  decimal  point  in 
the  quotient  as  soon  as  the  mark  in 
the  dividend  is  reached. 


Divide : 


1. 
2. 


.32. 
.046. 
.067. 


21.76   by 

4.462  by 

.1005  by 

.4984  by  5.6. 

.051  by   .85. 

395.52  by   .0309. 

3495.9  by  .0215. 

9.7696  by  .172. 

.16854  by  .00795. 

2280.96  by   .0324. 

complex  decimal  is  handled  in 
division  nearly  the  same  as  in  mixed 
numbers. 

Divide  4.678^^  by  .08. 

58.44%, 


4. 
5. 
6. 
7. 
8. 
9. 
10. 
A 


.08)4.67'8  ^n 

(Side  Work:) 


V7  =  ^Vl- 


/ 1 


=  ^% 


'56- 


Divide : 


1.  4.8973%  by   .7. 

2.  .3064%'!  by  .009. 

3.  .006395^  by  15. 

4.  .00429^  by  600. 

5.  .30973/^  by  2.7. 

6.  265.384%  by  7000. 

If  the  dividend  of  a  complex  deci- 
mal does  not  contain  as  many  deci- 
mal places  as  the  divisor  has,  carry 
the  common  fraction  out  as  many 
additional  decimal  places  as  are  de- 
sired. 

Divide  4.83%2  by  .0017. 
5/i2=  .412/3,  hence 
4. 83%i2  =  4.8341%. 


28433%! 

.0017)4.8341'% 
3  4 


1  43 

1  36 


74 
68 

61 
51 

10% 


Divide ; 


(Side   Work) 
10%  =  3% 

3%--17=32/^j 


1.  76. 5%  1  by   .023. 

2.  .07%4  by   .7365. 

3.  83^  by  6.295. 

4.  24.0%  by  .0034. 

If  the  divisor  is  a  complex  deci- 
mal reduce  both  dividend  and  divisor 
to  fractions  of  the  same  denomina- 
tor. 

Divide    .6795    by   2.43%. 
2.43%         .6795 
X9  X9 


21.89  6.1155 

972052/ 

21. 89)  6.  ir  55 
4  378 


1  7375 
1  5323 


2052 
Divide    .0792%   by    .174%. 
.174J^       .0792% 
X9  X9 


1.569 


.7133 

.45725^.^3 


1.569).713'30 
627  6 


85  70 
78  45 

7  25 


66 


ARITHMETIC 


Divide 

1.  6.573  by   .073/7. 

2.  84.3  by   .29^. 

3.  .573  by  2.8^11. 

4.  .643%!  by  4-/„. 

5.  1.2195%  by  2.3%. 

6.  6.3%2  by   .361,^-6. 

If  there  is  a  remainder  after  the 
quotient  has  been  carried  the  desired 
number  of  decimal  places  the  result 
may  be  written  as  a  complex  deci- 
mal, the  remainder  may  he  dropped, 
or  it  may  be  noted  and  retained, 
care  being  taken  to  place  the  deci- 
mal point  properly. 

Divide  47.6  by  370,   carrying  the 
result    to    three    decimal    places    and 
giving  the  remainder. 
.1  28 


37^)47.60 
37 

10  6 

7  4 

3  20 
2  96 

.24   Remainder 
In   the   following    examples    carry 
the    result    to    three    decimal    places 
and  state  what  the  remainder  is. 
Divide 

1.  .027  by  5.6. 

2.  7.6  by    .014. 

3.  .0068  by    .235. 

4.  900  by  .0013. 

5.  .37  by  170. 

6.  29.5  by   .093. 

Miscellaneous  Exercises  in  Division. 

Divide 

1.  100  by   .001. 

2.  .0003  by  3000. 

3.  3240  by   .027. 

4.  .00796  by  500. 

5.  .96064  by  .32. 

6.  425.92  by  .605. 

7.  3.4356  by  40.9. 

8.  9101.57  by  .0007. 


9.  6660  by   .074. 

10.  6.1472  by  6.8. 

11.  4.67yi4  bv   .33. 

12.  2.78142  by  3.07. 

13.  .265  by  6.7%. 

14.  2.322  bv  86. 

15.  .0003  by  1. 

16.  .0022  by  200. 

FRACTIONAL  PAHTS. 

(1).  To  find  ^  of  a  number,  di- 
vide it  by  4  to  get  ^  of  it,  and  mul- 
tiply the  quotient  by  3. 

Thus  to  find  }i  of  24. 

>4  of  24=0. 

Then  3/4  of  24=18. 

Other  fractional  parts  of  numbers 
may  be   found   in   a   similar   manner. 

Find  ■}'■;  of  5G. 

1/1  of  56=8. 

Then  ■}'■;  of  56=40. 

Find: 


1. 
2. 
3. 
4. 
5. 


%  of  63. 
74  of  32. 
%  of  84. 
%  of  126. 

%  of  36. 


6. 
7. 
8. 
9. 


%  of  70. 
54  of  50. 
%  of  21. 
H  of  42. 


This  is  virtually  multiplying  the 
number  by  the  fraction  denoting  the 
part  to  be  found. 

Find  %  of  84. 

%  of  84=72  or  64X84=72. 

Find  by  the  last  process: 


1. 
2. 
3. 
4. 
5. 


%  of  54. 
~Ai  of  88. 
5/10  of  36. 
%  of  14. 
%  of  30. 


6. 

7. 
8. 
9. 


%  of  67. 
%  of  Yu. 

r-  of  51/4- 
m   of  7%. 


(2).      Find    a    number    %    larger 
than  24.     To  do  this  find  %  of  the 
number    (24    in    this    example)     and 
add  it  to  the  number  itself. 
%  of  24=16. 
24 -f  16=40  Ans. 
Find  a  number: 

1.  Yz  larger  than  65. 

2.  %  larger  than  42. 

3.  %  larger  than   63. 

4.  ^%  larger  than  56. 

5.  %  larger  than  35. 

6.  %  larger  than  245%. 


FRACTIONAL   PARTS 


67 


This  class  of  problems  may  also  be 
solved  as  follows : 

Find  a  number  Yj  larger  than  63. 

A  number  is  once  itself.  Once  a 
number  plus  ^  of  it=i^  of  it. 
11/7  of  63=99. 

Find  a  number: 

1.  %  larger  than  72. 

2.  %2  larger  than  30. 

3.  %  larger  than  27. 

4.  %  larger  than  28. 

5.  %  larger  than  ^%5. 

6.  Yi  larger  than  3%. 

(3).  Find  a  number  ^  smaller 
than  45. 

1/5  of  45=9. 
45—9=36. 
Find  a  number: 

1.  %  smaller  than  49. 

2.  %i  smaller  than  66. 

3.  %  smaller  than  54. 

4.  %  smaller  than  18. 

5.  %  smaller  than  %. 

6.  Yq  smaller  than  46^. 
Problems    like   these   may   also   be 

solved  as   follows : 

Find  a  number  %2  less  than  48 
A  number  is  once  itself.  Once  a 
number  minus  %2  oi  it  leave?  yi2 
of  it. 

V12  of  48  =  28. 

Find  a  number 

1.  %  smaller  than  77. 

2.  %  less  than  40. 

3.  %5  less  than  120. 

4.  %  less  than  60. 

5.  %  less  than  %. 

6.  Ys  less  than  22%. 

PROBLEMS  INVOLVING  FRACTIONAL 
PARTS. 

1.  Henry  has  56  marbles  and 
James  has  %  as  many.  How  many 
has  James  ? 

2.  Mr.  Mason  had  42  tons  of 
dried  prunes  in  1912.  Tn  1913  his 
crop  was  %  as  large.  What  was  his 
crop  in  1913? 


3.  He  sold  his  prunes  at  $70  a 
ton  in  1912,  and  for  yi  more  per 
ton  in  1913.  How  much  did  he  re- 
ceive for  each  crop? 

4.  Raymond  is  5  feet  tall,  Sher- 
man is  Y12  taller,  and  Homer  is  % 
as  tall  as  Sherman.  Plow  tall  are 
Sherman  and  Homer,  respectively? 
(Answer   in    feet   and    inches). 

5.  An  apple  tree  bore  720  pounds 
of  fruit  and  the  crop  on  a  peach 
tree  was  %  lighter.  A  cherry  tree 
had  a  crop  Ys  heavier  than  that  on 
the  peach  tree.  How  many  pounds 
did  the  cherry  tree  bear? 

6.  A  hog  is  worth  $20.  If  a 
sheep  is  worth  %o  as  much  as  a 
hog,  and  a  goat  is  worth  %  as  much 
as  a  sheep,  what  is  the  value  of  75 
hogs,  350  sheep  and  12  goats? 

7.  A  boat  can  run  24  miles  an 
hour,  a  passenger  train  can  run  % 
faster,  and  an  aeroplane  can  travel 
%  faster  than  the  passenger  train. 
How  long  will  it  take  each  to  travel 
1200  miles? 

8.  A  cubic  foot  of  fresh  water 
weighs  1000  ounces  and  sea  water 
is  Y40  heavier.  Find  the  weight  in 
pounds  of  a  cubic  yard  of  sea  wa- 
ter. 

9.  Cork  is  %  lighter  than  fresh 
water.  Find  the  weight  in  pounds 
of  50  cubic  feet  of  cork. 

10.  A  tree  9  feet  tall  increased 
its  height  Y?.  each  year  for  four 
years.  What  was  its  height  at  the 
end  of  the  fourth  year? 

(4).     35  is  %  of  what  number? 
If  %  of  the  number=35 
Yr  of  the  number=  7 
%  of  the  number=49 

1.  48  is  %   of  what  number? 

2.  36  is   %   of  what   number? 

3.  24  is  %  of  what  number? 

4.  54   is   %   of   that   number? 

5.  231  is  %  of  what  number? 

6.  15  is  54  of  what  number? 

7.  25   is  %  of  what  number? 

8.  17^3  is  ^1  of  what  number? 

9.  8%  is  %  of  what  number? 
10.     %  is  %  of  what  number? 


68 


ARITHMETIC 


The  same  result  will  be  obtained 
in  examples  like  those  just  given  by 
dividing  the  number  by  the  fraction 
representing  the  part. 

105  is  %  of  what  number? 
105^%=105  X  %=75. 

1.  30    is    %  of  what  number? 

2.  75  is  %  of  what  number? 

3.  144  is  1%  of  what  number? 

4.  175  IS  %  of  what  number? 

5.  27  is  %  of  what  number? 

6.  65  is  %3  of  what  number? 

7.  S%  is  %  of  what  number? 

8.  i%4  is  %  of  what  number? 

(5).  30  is  %  greater  than  what 
number?  A  number  is  %  of  itself. 
Once  a  number  plus  Y^  of  it  equals 
%  of  it. 

%  of  the  number=30 
%  of  the  number=5 

%  of  the  number=25     Ans. 

1.  84  is  Yq  greater  than  what  no.  r 

2.  45  is  %  greater  than  what  no.? 

3.  56  is  %  greater  than  what  no.  ? 

4.  120  is  %  greater  than  what  no.? 

5.  18  is  Yd  greater  than  what  no.  ? 

6.  8%  is  Yi  greater  than  what  no.? 

7.  5Y2  i''  %  greater  than  what  no,  ? 

8.  ■^%5  is  Yo  greater  than  what  no.? 

The  work  may  be  shortened  by 
division.  120  is  %  greater  than 
what  number? 

5/13/ — 8^ 
/5   I    /5 /5 

%  of  the  number  =  120. 

120-^%=    75    Ans. 

1.  126  is  %  greater  than  what  no.? 

2.  264  is  %  greater  than  what  no.? 

3.  560  is  %  greater  than  what  no.? 

4.  585  is  %  greater  than  what  no.? 

5.  180  is  %  greater  than  what  no.? 

6.  26%  is  Yti  greater  than  what  no.  ? 

7.  1034  is  %  greater  than  what  no.? 

8.  ^%5  is  %  greater  than  what  no.  rf7 

(6).  56  is  %  less  than  what  num- 
ber?    A  number  is  %  of  itself,  and 


if  %  of  it  is  taken  away,  '^  will  be 
left. 

^4  of  the  number=56. 
Yr  of  the  number=14. 
%  of  the  number=98  Ans. 

1.  18    is    Ys    less    than    what  no.? 

2.  120    is    %    less    than    what  no.? 

3.  252    is    %    less    than    what  no? 

4.  84    is    Yi    less    than    what  no.f 

5.  7%    is    %    less    than    what  no.? 

6.  ^%o    is    ii    less    than    what  no.? 

The  process  may  be  shortened  by 
division.  75  is  %  less  than  what 
number  ? 

%  of  the  number=  75. 
75-^-%=125  Ans. 

1.  126    is    %    less    than  what  no.? 

2.  48    is    Yi    less    than  what  no.? 

3.  36    is    %    less    than  what  no.  ? 

4.  25    is    %    less    than  what  no.? 

5.  5%    is    %    less    than  what  no.? 

6.  %    is    %    less    than  what  no.  ? 

PROBLEMS. 

1.  Mary  has  45  cents,  which  is 
%  of  what  Helen  has.  How  much 
has  Helen? 

2.  John  has  %  as  many  marbles 
as  Herbert  and  they  have  together 
56   marbles.     How   many   has   each? 

3.  Mr.  Morgan  raised  240  tons 
of  prunes  and  his  crop  was  %  larger 
than  Mr.  Payne's.  What  was  Mr. 
Payne's  crop? 

4.  The  distance  from  Palo  Alto 
to  San  Francisco  is  %  greater  than 
the  distance  from  San  Jose  to  Palo 
Alto  and  the  distance  from  San  Jose 
to  San  Francisco  is  51  miles.  What 
is  the  distance  from  San  Jose  to 
Palo  Alto? 

5.  An  orchard  consisting  of  peach 
trees  and  apricot  trees  contains  800 
trees.  There  are  %  more  peach 
trees  than  apricot  trees.  How  many 
of  each  kind? 


FRACTIONAL   PARTS 


69 


6.  Mary  is  42  inches  tall  and  she 
is  %i  shorter  than  Hannah.  How 
tall  is  Hannah? 

7.  An  oak  was  75  feet  tall.  It 
was  %  taller  than  a  walnut  and  the 
walnut  was  ^  shorter  than  an  eucal- 
yptus. How  tall  was  the  eucalyp- 
tus? 

(7).  18  is  what  part  of  24? 
1  is  1/24  of  24. 
18  is  i%4  of  24. 
^%4=%-   Hence  18  is  %  of  24. 

As  a  rule  the  result  is  obtained  di- 
rectly by  writing  the  result  in  frac- 
tional form  at  once  and  reducing  to 
its  lowest  terms. 

i%4=r3^.     Hence  18  is  %  of  24. 

28  is  what  part  of  35? 
2%5=^.     Hence  28  is  ji  of  35. 

42  is  what  part  of  32? 

*%2=-yi6.  Hence  42  is  ^Y^e  of  32. 

What  part  of  18  is  27? 

2%g=3/2.     Hence  27   is  %  of  18. 

Notice  that  the  part,  which  be- 
comes the  numerator  of  the  fraction, 
is  the  subject  of  the  sentence,  and 
that  the  whole,  which  becomes  the 
denominator  of  the  fraction,  is  the 
object  of  the   preposition   of. 

1.  28  is  what  part  of  42? 

2.  36  is  what  part  of  63? 

3.  64  is  what  part  of  48? 

4.  What  part  of  16  is  26. 

5.  8  inches  is  what  part  of  3  feet? 

6.  125  rd,  is  what  part  of  a  mile? 

Fractions  are  handled  in  a  similar 
manner.     2}i  is  what  part  of  22? 

"22 788 V&- 

Hence  2^   is   Ys  of  22. 
%  is  what  part  of  5)^  ? 


■v^=r7X%6=%6     Ans. 

1.  4^  is  what  part  of  6%? 

2.  %  is  what  part  of  1^  ? 

3.  2/3   ft.  is  what  part  of  %  yd? 

4.  What  part  of  3^   is   1^? 

5.  What  part  of  1%  is  4%? 

6.  What  part  ol-  %  yd.  is  ^  ft.? 


(8).  63  is  what  part  greater  than 
36? 

It  is  necessary  to  find  how  much 
it  is  greater. 

63—36=27,  ^y3Q=H-  Hence  63 
is  %  greater  than  36. 

54  is  what  part  greater  than  42? 

54—42=12,  ^%2=y7,  hence  54  is 
%  greater  than  42. 

Notice  that  the  difference  between 
the  two  numbers  is  the  part  and  be- 
comes the  numerator  of  the  fraction, 
and  that  the  whole,  which  becomes 
the  denominator  of  the  fraction,  fol- 
lows than. 

75  is  what  part  greater  than  45? 

75-^5=30,  3%5=^.  Hence  75 
is   ^   greater  than  45. 

1.  60  is  what  part  greater  than 
42? 

2.  75  is  what  part  greater  than 
54? 

3.  180  is  what  part  greater  than 
80? 

4.  i%6  is  what  part  greater  than 

^2? 

5.  9^  is  what  part  greater  than 
5>4? 

6.  %    is   what   part   greater   than 
11 

(9).  32  is  what  part  less  than 
44? 

It  is  necessary  to  find  how  much 
32  is  less  than  44  first.  44—32=12 ; 
i%4=%i.  Hence  32  is  %i  less  than 
44. 

36  is  what  part  less  than  56? 

56  —  36  =  20,  2%6  =  %4.  Hence 
the  answer  is  %4. 

Notice  that  the  difference  between 
the  numbers  becomes  the  numerator 
of  the  fraction,  and  that  the  whole, 
which  becomes  the  denominator,  fol- 
lows the  word  than. 

64  is  what  part  less  than  92? 

92  —  64  =  28,  28/92  =  723,  Ans. 

1.  56  is  what  part  less  than  77? 

2.  45  is  what  part  less  than  75? 

3.  32  is  what  part  less  than  84? 

4.  23  is  what  part  less  than  31? 

5.  %   is  what  part  less  than  %? 

6.  3%  is  what  part  less  than  5%? 


Yi^' 


70 


ARITHMETIC 


mSCELLANEOTJS     EXERCISES       IN 
FRACTIONAL    PARTS. 

1.  Find  (1)  ^  of  84;  (2)  a 
number  %  greater  than  84;  (3)  a 
number  %  less  than  84, 

2.  (1)   84  is  ^  of  what  number? 

(2)  ^    greater  than   what   number? 

(3)  ^  less  than  what  number? 

3.  (1)  56  is  what  part  of  84? 
(2)  what  part  less  than  84?  (3) 
84  is  what  part  greater  than  56? 

4.  ^  of  84  is  ^  greater  than 
what  number? 

5.  %  of  84  is  %  less  than  what 
number  ? 

6.  Find  a  number  i%  larger  than 
126. 

7.  126  is  1%  larger  than  what 
number  ? 

8.  75  is  Ys  smaller  than  what 
number  ? 

9.  Find  a  number  ^  smaller  than 
75. 

10.  100  is  what  part  larger  than 
75? 

PROBLEMS  IN  FRACTIONAL  PARTS. 

1.  Mr.  Stone  bought  a  horse  for 
$120,  and  sold  it  for  %  of  its  cost. 
How  much  did  he  gain? 

2.  Mr.  Andrews  sold  a  house  for 
$4800,  which  was  %  of  its  cost. 
What  was  his  gain? 

3.  Mr.  Brown  sold  a  bicycle  for 
$24,  which  was  Yz  less  than  its 
cost.     What  was  his  loss? 

4.  A  pole  28  feet  long  was 
broken  so  that  the  length  of  one 
part  was  %  of  the  length  of  the 
other.     Find  length  of  each  part. 

Let  %  =  the  length  of  the  longer 
part. 
Then  %=rthe  length  of  the  shorter 
part. 
And  %  =  the  length  of  the  whole. 
%  of  longer  part=28  ft. 
14  of  longer  part=  4  ft, 
%  of  longer  part=16  ft.   longer  part. 
%  of  longer  part=12  ft,    shorter   part. 


5.  The  sum  of  two  numbers  is 
240  and  the  smaller  is  ^  of  the 
larger.     Find  each  number. 

6.  The  difference  between  two 
numbers  is  84,  and  the  smaller  is  % 
of  the  larger.     Find  each  number. 

7.  The  sum  of  three  numbers  is 
132.  The  first  is  ^  of  the  second, 
and  the  third  is  Ys  of  the  first.  Find 
each  mimber. 

8.  Yi  of  Mary's  age  equals  Yo  of 
Horace's,  and  the  sum  of  their  ages 
is  24  years.     How  old  is  each? 

Since  %  of  Mary's  age=%  of 
Horace's. 

%  of  Mary's  age=%  of 
Horace's. 

And  %  of  Mary's  age=%  of 
Horace's. 

Let  %=Horace's  age. 
Then  %=Mary's  age. 
And  %=the  sum  of  their  ages. 
Then  %  of  Horace's  age=^24  years 
y^  of  Horace's  age=  3  years 
%  of  Horace's  age^  9  years 

Horace's  age. 
%  of  Horace's  age^l5  years 

Mary's  age. 

9.  An  estate  of  $27940  is  to  be 
shared  by  a  brother  and  sister  so 
that  Yz  of  the  brother's  share  is 
equal  to  ^  of  the  sister's.  Find  the 
share  of  each. 

10.  Mary  has  48  inches  of  ribbon. 
She  has  Y?,  more  than  Susan,  and 
Susan  has  %  more  than  Jane.  How 
many  inches  have  Susan  and  Jane 
respectively  ? 

Ans.,  S.  36  in.:  J.  28  in. 

11.  There  are  124  more  boys  than 
girls  at  a  certain  school,  and  % 
of  the  boys  equals  %  of  the  girls. 
How    many    students    are   there? 

12.  A  certain  orchard  consisting 
of  apples,  peaches  and  pear  trees 
contains  3600  trees;  Yz  of  the  apple 
trees  equals  Ya  of  the  peach  trees, 
and  Ya  of  the  peach  trees  equals  Y^ 
of  the  pear  trees.  How  many  trees 
of  each  kind  does  the  orchard  con- 
tain? 


PERCENTAGE 


71 


13.  Hannah  is  60  inches  tall.  She 
is  yi  shorter  than  John,  and  John  is 
y^  taller  than  Emily.  How  tall  is 
Emily?  Ans.,  54  in. 

PERCENTAGE. 
(1)  To  Express  Per  Cent  as  a  Decimal. 

Per  cent  means  hundredths.  If 
the  per  cent  contains  no  decimal  or 
fractional  part,  its  is  only  necessary 
to  point  off  two  decimal  places  and 
omit  the  character  (%),  or  word 
per  cent. 

21%  =  21;  3  per  cent  =  .03; 
285%  =  2.85 
Express    the    following    decimally: 

1.  7%. 

2.  65%. 

3.  495%. 

4.  2  per  cent. 

5.  4000  per  cent. 

6.  967  per  cent. 

7.  800%. 

8.  40%. 

9.  100%. 

10.  5384%. 

11.  478%. 

12.  198%. 

If  the  per  cent  contains  a  decimal 
part  move  the  decimal  point  two 
places  to  the  left  and  omit  the  char- 
acter. 

Express  decimally: 

1.  .6%. 

2.  4.7%. 

3.  .25%. 

4.  46.5%. 

5.  895.37%. 

6.  .003%. 

7.  .09  per  cent. 

8.  4.9  per  cent. 

9.  .0079  per  cent. 

If  the  per  cent  contains  a  com- 
mon fraction  express  it  as  a  com- 
plex decimal  and  reduce  the  result- 
ing expression  to  a  pure  or  mixed 
decimal,  if  possible. 

5i^%=.  05  >^=.  05125. 

3/^%=. 003/7. 


Express  decimally : 


1. 

H%- 

7. 

.73/40%. 

2. 

y25%. 

8. 

•0%25%. 

3. 

43^%. 

9. 

200^%. 

4. 

M%. 

10. 

•6%!%. 

5. 

6yi6%. 

11. 

268>^%. 

6. 

2471/50%. 

12. 

.18%%. 

(2)  To  Express  Per  Cent  as  a  Common 
Fraction. 

In  general  it  is  best  to  change 
the  per  cent  to  a  decimal  and  then 
change  that  result  to  a  common  frac- 
tion and  reduce  to  its  lowest  terms. 

15%=.15=i%oo=%o; 

Express  as  a  common  fraction  in 
its   lowest  terms : 

1.  48%.  7.  260%. 

2.  225%.  8.  71/7%. 

3.  4.5%.  9.  .0%%. 

4.  .025%.  10.  33>^%. 

5.  .%%%.  11.  .33^6%. 

6.  .055/7%.  12.  3.3>^%. 
The    work    may  be    shortened    by 

omitting  the  decimal  form,  being 
careful  to  annex  the  two  ciphers  to 
the  denominator  in  place  of  the  char- 
acter,  %. 

.24%==2%0000=%250; 

93/7%=66/^00=^%50; 

1.4y6%=«%000=l%200. 

Express  as  a  common  fraction  in 
its  lowest  terms: 

1.  13^%.  6.  4.5%i%. 

2.  .6^%.  7.  .571/7%. 

3.  .05/7%.  8.  76^%. 

4.  71/7%.  9.  400^%. 

5.  221/9%.  10.  5/J%. 
Some    per    cents  freuqently    used 

reduce  to  small  fractions.  Pupils 
should  become  so  familiar  with  these 
that  they  will  readily  recognize  them. 
For  example,  25%=i4 ;  50%=^; 
75%=^. 

Reduce  to  common   fractions : 

1.  10%,  20%,  30%,  40%,  50%, 
60%,  70%,  80%,  90%. 

2.  8>^%,  16^%,  3334%, 66^%.- 


72 


ARITHMETIC 


3.  Count  by  12>^  to  100,  begin- 
ning with  12>4.  Write  the  character 
%  after  each  result  and  reduce  to  a 
common  fraction  in  its  lowest  terms. 

4.  Count  by  8>^  from  Sys  to 
100,  annexing  the  character  %  after 
each  result  and  reduce  as  in  exer- 
cise 3. 

5.  Count  by  11%  to  100  and  treat 
the   results   as   in   exercise   3. 

6.  Count  by  14%  to  100  and  treat 
the   results   as   in   exercise   3. 

(3)   To  Find  a  Given  Per  Cent  of  a 
Number. 

As  a  rule  it  is  best  to  express  the 
per  cent  as   a   decimal  and  multiply 
the  given  number  by  it. 
Find  17%  of  7658. 

7658    . 
.17 


53606 
7658 


1301.86  Ans. 


Find  7^%  of  47.85. 
47.85 
.076 


28710 
3  3495 


3.63660 
Find  6^4%  of  375. 
375 


7)1500 


214% 
2250 


24.64% 
Find  by  the  method  here  used: 

1.  y3%  of  867. 

2.  128%  of  42.5. 
800%  of  375. 
25%   of  6.96. 
14%  of  7651. 
.8%%  of  7965. 

If  the  per  cent  reduces  to  a  small 
fraction  it  is  usually  best  to  use  the 
fraction. 


3. 
4. 
5. 
6. 


Find  16^%  of  894.     16^%=i^  ; 
;  of  894=149  Ans. 

Find  621/0%  of  6.256.     62>^  =  ^  ; 
5  of  6.256=3.910  Ans. 

Find  by  using  fractions : 

1. 

2. 

3. 

4. 

5. 

6. 


75%  of  788. 
8Sy3%  of  35.4. 
36fii%   of  4.632. 
201/7%  of  238. 
871^%  of  38. 
77%%  of  127%. 


Find  by  the  easiest  method: 

1.  %%  of  6545. 

2.  ^%   of  6752. 

3.  12y.%  of  47.31. 

4.  150%  of  ys. 

5.  V5%  of  4%. 

6.  4%%  of  1.65%. 

(4).    To  find  a  number  a  given  per 
cent  larger  than  a  given  number. 


To  do  this  it  is 
find  the  per  cent  ( 
add  this  to  the  nu 
Find  a  number 
486. 

15%=.  15. 
486 
.15 

only 
3f  the 
mber. 

15% 

is  10 

Vc. 

necessary  to 
:  number  and 

greater  than 

486 
72.90 

2430 
486 

558.90   Ans 

72.90 
or  since  a  number 
100%+15%=115^ 
4  86 
1.15 

i0%  of  itself, 

24  30 

48  6 
486 


558.90    Ans. 

number    16^3% 


greater 


Find    a 
than  593. 

16i^%=K;  %  of  593=985/^. 
593  plus  98^=691%  Ans. 

In  what  follows  use  whatever  plan 
seems  shortest. 


PERCENTAGE 


73 


Find  a  number: 

1.     17%   larger  than  27. 

275%   larger  than  654. 

2^%  larger  than  1440. 

66%%  larger  than  216. 

Yifo  larger  than  945. 


ys%  larger  than  674. 


(5).  To  find  a  number  a  given  per 
cent  less  than  a  given  number. 

The  process  is  identical  with  that 
of  (4)  except  that  after  the  required 
per  cent  is  found,  decimally  or  oth- 
erwise, it  is  subtracted  from  the  or- 
iginal number. 

Find  a  number  24%  less  than  575. 
24%=.  24. 

575  575 

.24  138 


2300 
1150 

=  76%: 

437 
=  .76, 

Ans 

or 

138.000 
100%— 24%  = 
575 

.76 

3450 
4025 

437.00  Ans. 

Find  a  number  37>^%  less  than 
645. 

S7V2%=H;  H  of  645=241^. 

645—241 7/^=403 %  Ans. 

100%— 37i^%=62^%=H ; 

^  of  645=403>^  Ans. 

In  what  follows,  use  the  method 
which   seems   shortest. 

Find  a  number: 

1.  26%  less  than  182. 

2.  3.7%  less  than  579. 

3.  %%  less  than  837. 

4.  333/3%  less  than  548. 
Gy3%  less  than  759. 

less  than  2681. 


6. 


■0%% 


Problems. 


1.      Sea    water    is    2.7%    heavier 
than    fresh    water,   and   a   cubic   foot 


of  fresh  water  weighs  62%  pounds. 
Find  the  weight  of  a  cubic  yard  of 
sea  water. 

2.  Ice  is  7%  lighter  than  fresh 
water.  Find  the  weight  of  100  cu- 
ft.  of  ice. 

3.  An  orchard  contains  600  peach 
trees.  There  are  60%  more  prune 
trees  than  peach  trees;  60%  fewer 
apple  trees  than  peach  trees,  and 
the  number  of  pear  trees  is  60%  of 
the  number  of  peach  trees.  How 
many  trees  does  the  orchard  contain? 

4.  Mrs.  Kammerer  put  1200  eggs 
in  an  incubator.  95%  of  the  eggs 
hatched  and  60%  of  the  chicks  were 
pullets.  She  sold  75%  of  the  roost- 
ers.     How    many    chickens    remain? 

5.  A  tree  8  feet  tall  increased  its 
height  25%  each  year  for  4  years. 
What  was  its  height  at  the  end  of 
the  fourth  year? 

6.  Mr.  Burton  bought  a  house  for 
$4500  and  afterward  sold  it  at  an 
advance  of  10%  on  its  cost.  The 
purchaser  sold  it  at  an  advance  of 
10%  and  the  last  purchaser  sold  at 
a  reduction  of  20%.  What  was  the 
last  selling  price? 

7.  A  city  whose  population  was 
1200  in  1900  increased  in  popula- 
tion 275%  in  ten  years.  At  this  rate 
of  increase,  what  will  its  population 
be  in  1920? 

8.  2.55%  of  the  weight  of  sea 
water  is  salt.  How  much  salt  can 
be  obtained  from  16000  cubic  yards 
of  sea  water? 

9.  The  waters  of  Salt  Lake  are 
20%  heavier  than  fresh  water  and 
13%  of  this  weight  is  common  salt. 
The  area  of  the  lake  is  2500"  sq. 
miles,  and  its  average  depth  is  20 
ft.  How  many  tons  of  salt  does  the 
lake  contain? 

(6)  To  find  a  number  when  a 
certain  per  cent  of  it  is  given. 

In  finding  a  given  per  cent  of  a 
number  the  number  is  multiplied  by 


74 


ARITHMETIC 


the  per  cent  expressed   decimally  or 
as   a  common   fraction. 

The  per  cent  of  a  number,  then, 
is  the  product  and  the  number  and 
the  per  cent  are  factors.  Flence  to 
find  a  number  when  a  certain  per 
cent  of  it  is  given,  it  is  only  neces- 
sary to  divide  the  given  per  cent  of 
the  number  by  the  per  cent  expressed 
either  decimally  or  as  a  fraction, 

834  is  15%  of  what  number? 

15%=.  15. 

55  60 


.15)834.00 
826  is  87>4%  of  what  number? 
87>^%  =  %. 
Ys  of  the  number=826. 
^  of  the  number=119. 
%  of  the  number=952. 
(or  826^J^=952.) 

In  the  following  exercises  express 
the  per  cent  decimally: 

1.  731  is  17%   of  what  number? 

2.  378  is  135%  of  what  number? 

3.  1045  is  8K%  of  what  number? 

4.  684  is  4%7o  of  what  number? 

5.  12.25  is  .3%%  of  what  number? 

6.  8715  is  275%  of  what  number? 

In  the  following  problems  express 
the  per  cent  as  a  fraction : 

L  483  is  87>^%  of  what  num- 
ber? 

2.  895  is  16^%  of  what  num- 
ber ? 

3.  764  is  44%%  of  what  num- 
ber? 

4.  287  is  37>4%  of  what  number? 

5.  5.754  is  162>^%  of  what  num- 
ber? 

6.  687  is   yz%  of  what  number? 

(7).  To  find  a  number  when  a 
certain  per  cent  more  than  it  is  giv- 
en. 

Since  any  number  is  once  itself, 
or  100%  of  itself,  it  is  only  neces- 
sary' to  add  the  given  per  cent,  ex- 
pressed either  decimally  or  as  a 
fraction  to  1  similarly  expressed,  and 
divide  the  given  number  Ijy  this  sum. 


729    is    35%    greater     than     what 
number  ? 

100%-f  35%=135% ; 
135%=1.35. 

5  40     Ans. 


1.35)729.00' 
675 


54  0 
54  0 


594  is  37>4%  greater  than  what 
number  ? 

37>^%=^;  1+^=11/8.  lyg  of 
number=594.     594-f-ii^=432  Aps. 

In  the  following  use  whichever 
method  seems  shortest: 

1.  996  is  335^%  more  than  what 
number  ? 

2.  1764  is  22y2%  more  than  what 
number  ? 

3.  29.07  is  112y2%  more  than 
what  number? 

4.  1.397  is  46^^%  more  than 
what  number? 

5.  601.5  IS,  %%  more  than  what 
number  ? 

6.  28105  is  ]4i%  more  than  what 
number  ? 

(8).  To  find  a  number  when  a 
certain  per  cent  less  than  it  is  given. 

Subtract  given  per  cent  from 
100%  and  expressing  the  given  re- 
sult decimally,  divide  the  given  num- 
ber by  it. 

396  is  45%  less  than  what  num- 
ber? 

100%— 45%=55%  ;  55%=. 55. 
7  20  Ans 


,55)396.00' 
385 


11  0 
11  0 


56.73    is    57%%    less    than    what 
number  ? 

100%— 57i/7%=426/7%  ; 

42%%=-%. 

%  of  number=56.73. 

56.73-f-%=132.37. 


PERCENTAGE 


75 


Use  the  method  that  seems  short- 
est in  the  following  examples. 

1.  103  is  15%  less  than  what 
number  ? 

2.  894  is  3Zy3%  less  than  what 
number  ? 

3.  4695  is  37>^%  less  than  what 
number  ? 

4.  %Q  is  62y2%  less  than  what 
number  ? 

5.  173.92  is  2%%  less  than  what 
number  ? 

6.  156.6  is  33^%  less  than  what 
dumber  ? 

Problems. 

1.  1173  is  15%  of  what  number? 
15%  greater  than  what  number? 
15%   less  than  what  number? 

2.  Pine  wood  is  55%  lighter  than 
water.  How  many  cubic  feet  in  a 
ton  of  pine  wood? 

3.  What  is  the  weight  of  1000 
feet  of  pine  lumber? 

4.  Mr.  Reynolds  had  25%  more 
prunes  than  Mr.  Leonard,  and  Mr. 
Woodham  had  10%  more  than  Mr. 
Reynolds.  They  had  together  3996 
tons.     How  much   had  each? 

5.  Mr.  Munroe  left  $15290  to  his 
three  children,  Henry,  Mary  and 
Susan.  He  gave  Mary  30%  more 
than  Henry  and  Susan  12^%  less 
than  Mary.  How  much  did  he  give 
"each? 

6.  Milk  is  3.2%  heavier  than  wa- 
ter. How  much  should  a  gallon  (231 
cu.  in.)   of  milk  weigh? 

7.  A  tree  increased  its  height 
20%  each  year  for  four  years  and  was 
then  54  feet  tall.  How  tall  was  it  at 
first? 

8.  Mrs.  Stewart  is  12%  shorter 
than  her  husband  and  the  sum  of 
their .  height  is  117^2  inches.  Find 
the  height  of  each. 

9.  Mr.  Simpson  sold  a  horse  los- 
ing 10%  of  its  cost.  With  the  mon- 
ey he  bought  another  which  he  sold 
at  a  loss  of  10%.  His  total  loss  was 
$23.75.  What  did  the  first  horse  cost  ? 

10.  Mr.  Johnson  sold  a  house  at  a 
loss   of  30%   on   the   cost:   with   the 


money  he  bought  another  which  he 
sold  at  a  gain  of  30%  on  its  cost. 
His  net  loss  was  $432.  Find  the 
cost  of  each  house. 

(9).  To  express  a  decimal  as  per 
cent. 

Since  per  cent  means  hundredths 
it  is  only  necessary,  if  the  number 
consists  of  two  decimal  places,  J:o 
omit  the  decimal  point  and  use  the 
character  %. 

Thus,  .  17=17 %  ;  .  0314=31/7 %  ; 
3.67=367%. 

If  the  expression  contains  less 
than  two  decimal  places,  extend  it  to 
two  places,  omit  the  point,  and  use 
the  character  as  before. 

Thus,  3.7=3.70=370%,, 

.63/ii=.62%i=628/ii%. 

4^=4.60=460%,. 

.7%5=.70%7=70%7%v 

If  there  are  more  than  two  deci- 
mal places  in  the  expression  move 
the  decimal  point  two  places  to  the 
right  and  introduce  the  character. 

Thus,   .364=36.4%  or  362/5%; 

. 0475=4. 75%o  or  4%%?; 

.007%=, 7%  or  i3/^g%. 

Express  the  following  as  per  cent: 

1.  2,042/7.  9.  3/050. 

2.  20,42/7.  10.  .0007/250- 

3.  2042/7.  11,  ,0%5o. 

4.  ,204%.  12.  .OOyaso. 

5.  ,83X25.  13.  .0025. 

6.  83/125,  14.  225. 

7.  .008%25-  15.  .1225. 

8.  .083/123.  16.  12.25. 

(10).  To  express  a  common  frac- 
tion as  per  cent. 

It  is  only  necessary  to  reduce  the 
fraction  to  a  decimal  of  at  least  two 
decimal  places  and  then  use  the  char- 
acter %  instead  of  two  decimal  plac- 
es. 

Thus,   5/i8=.27%=27%%. 

%25=.0064=.64%. 

8i4o=2. 025=202, 5%o. 


76 


ARITHMETIC 


Express  the  following  fractions  as 
per  cent: 


1. 

"/25. 

9. 

.01/7. 

2. 

i«%o. 

10. 

.ooy. 

3. 

4yi6. 

11. 

Vi. 

4. 

.9y4. 

12. 

.0001/7, 

5. 

3.7>^. 

13. 

.503/49. 

6. 

%7. 

14. 

5.03/49, 

7. 

49%5. 

15. 

.503/40. 

8. 

7.0%4. 

16. 

.0%. 

(11).  To  find  what  per  cent  one 
number  is  of  another. 

It  will  be  seen  that  this  is  similar 
to  (7)  of  fractional  parts.  First  ex- 
press the  relation  in  the  form  of  a 
fraction,  and  then  change  the  frac- 
tion to  per  cent. 

15  is  what  per  cent  of  48? 

i%8=. 3125=31. 25%  or  31i/4%. 


27 

is  what  per  cent  of  67? 

2%7=-.4020/6,=  4020/e„%     Ans, 

.4020/,, 

67)27.00 

26  8 

20 

Find  what  per  cent: 

1. 

65  is  of  125. 

2. 

88  is  of  160. 

3. 

629  is  of  125. 

4. 

43%  is  of  19. 

5. 

84  is  of  217%,. 

6. 

7  is  of  1250. 

7. 

41/12  is  of  161^. 

8. 

593/  is  of  84%. 

9. 

6.75  is  of  27.5. 

10. 

7.4  is  of  2.75. 

11. 

.674  is  of  25. 

12. 

56.4  is  of  25. 

(12).  -To  find  what  per  cent  one 
number  is  greater  than  another. 

This  is  similar  to  (8)  in  fraction- 
al parts.  First  find  how  much  one 
number  is  greater  than  the  other, 
then  find  what  per  cent  this  differ- 
ence is  of  the  other  number. 


87  is  what  per  cent  greater  than 
75? 


87—75=12. 


.16 


75)12.00 

7  5 
7  5 


4  50 
4  50 


.16=16%  Ans. 

478  is  what  per  cent  greater  than 
58? 

478  —  58  =  420 ; 

7.24y29 


58)420.00 
406 


14  0 
11  6 

2  40 
2  32 


8 

%8=r29. 

7.24y29=724%9%  Ans. 

%   is   what  per  cent   greater   than 

%?    %-%='A^\ 

y3C-^%=y2o=35%   Ans. 
Find  what  per  cent: 

1.  49  is  greater  than  40? 

2.  88  is  greater  than  30? 

3.  126  is  greater  than  125? 

4.  %  is  greater  than  3^^? 

5.  4%  is  greater  than  3%? 

6.  76.4  is  greater  than  7.64? 

(13).  To  find  what  per  cent  one 
number  is  less  than  another? 

This  is  similar  to  (9)  in  fraction- 
al parts.  First  find  how  much  it  is 
less  than  the  other  number,  and  then 
divide  this  difference  by  that  num- 
ber. 

45  is  what  per  cent  less  than  60? 

60 — 45=15 

.2  5 


60) r 5.0 
,25=25%   Ans. 


PERCENTAGE 


77 


2.15    is    what    per    cent    less   than 
3.6? 

3.6—2.15=1.45 


3.6)].4'50 
14  4 


10 

.40^i8=40%,s%  Ans. 
Find  what  per  cent : 

1.  147  is  less  than  250? 

2.  265  is  less  than  625? 

3.  .0475  is  less  than    .475? 

4.  3%  is  less  than  6}i? 

5.  %  is  less  than  %? 

6.  .6%  is  less  than  4%? 

In  solving  concrete  problems  the 
per  cent  should  always  be  connected 
with  some  object  which  is  the  base, 
or  100  per  cent.  For  example,  m 
problem  "3"  below  Mary's  age  is  the 
base. 

Solution. 

Let  100  per  cent  =  Mary's  age. 
Then  125  per  cent  =  John's  age 
and   225   per  cent  =  the   sum   of 
their  ages. 

225  per  cent  of  Mary's  age  equal 
36  years. 

Hence  Mary's  age  =  16  years, 
and  John's  age   =20  years. 

Per  cents  cannot  be  added  or  sub- 
tracted in  the  concrete  unless  they 
are  per  cents  of  the  same  thing.  4 
per  cent  of  a  yard,  5  per  cent  of  a 
foot,  and  6  per  cent  of  an  inch  can 
be  added  only  after  they  have  been 
changed  to  per  cents  of  the  same 
unit. 

4  per  cent  yd.  plus  5  per  cent  ft. 
plus  6  per  cent  in.  equal  12  per  cent 
ft,  plus  5  per  cent  ft.  plus  .5  per  cent 
ft.  equal  17.5  per  cent  ft. 

It  is  best  to  state  the  preliminary 
work  in  concrete  problems  fully  and 
explicitly.  The  solution  of  problem 
"5"  below  is  here  given  as  an  exam- 
ple. 

Solution. 

Let  100  per  cent  =  William's 
share. 


Then  120  per  cent  =  Samuel's 
share 

and  180  per  cent  =  Homer's 
share. 

400  per  cent  =  the  sum  of  shares. 

400  per  cent  of  William's  share 
=  $400. 

Hence  William's  share  =  $100. 

Samuel's  share  =  $120. 

Homer's  share  =  $180. 

PROBLEMS. 

Ij  A  number  which  is  20%  great- 
er than  ]80  is  25%  less  than  what 
number?     Ans.,  288. 

2.  The  number  which  is  25%  less 
than  180  is  25%  greater  than  what 
number?    Ans.,  108. 

3.  John  is  25%  older  than  Mary, 
and  the  sum  of  their  ages  is  36  years. 
How  old  is  each?  Ans.,  M.  16;  T. 
20. 

4.  Mary  is  40%  younger  than 
John  and  the  sum  of  their  ages  is 
40  years.  How  old  is  each?  Ans., 
M.  15;  J.  25. 

5.  Samuel  has  20%  more  money 
than  William,  and  Homer  has  50% 
more  than  Samuel.  They  have  to- 
gether $400.  How  much  has  each? 
Ans.,  W.  $100;  S.  $120;  H.  $180. 

6.  Mary  has  20%  more  chickens 
than  Susan,  and  Helen  has  33}^% 
fewer  than  Mary.  They  have  to- 
gether 360  chickens.  How  many  has 
each?  Ans.,  M.  144;  S.  120;  H.  96. 

7.  Mr.  Macy's  1909  crop  of  prunes 
was  20%  less  than  his  1908  crop, 
and  his  1910  crop  was  50%  more 
than  his  1909  crop.  His  1910  crop 
exceeded  his  1908  crop  by  6  tons. 
How  much  was  his  1910  crop?  Ans., 
36  tons. 

8.  Mr.  Bishop's  fruit  crop  was 
20%  less  than  Mr.  King's,  and  Mr. 
Ball's  crop  was  50%  more  than 
Mr.  Bishop's.  Mr.  Ball  had  7  tons 
more  than  Mr,  King.  How  much 
had  each?  Ans.,  K.  35  tons;  Bi.  28 
tons ;  Ba.  42  tons. 


78 


ARITHMETIC 


9.  If  cloth  loses  10%  of  its 
length  in  washing  and  dyeing,  how 
much  unwashed  and  undyed  cloth  is 
required  to  make  360  yards  of  cloth 
after  it  is  washed  and  dyed?  Ans., 
400  yards. 

10.  Mr.  Wise  bought  clothing  at 
$16  a  suit,  and  wishes  to  sell  it  at  a 
gain  of  15%.  How  should  he  mark 
it  so  that  he  may  reduce  the  price 
8%  and  still  gain  the  15%?  Ans., 
$20. 

11.  A  tree  increased  its  height 
50%  the  first  year,  was  then  cut  back 
20%,  and  the  second  year  increased 
its  height  33^^%,  when  its  full  height 
was  8  feet.  How  high  was  it  at 
first?     Ans.,  5  feet. 

12.  Mr.  Slocum  increased  his 
weight  20%,  then  lost  25%,  then 
gained  50%.  He  now  weighs  216 
lbs.  How  much  did  he  formerly 
weigh?     Ans.,  160  lbs. 

13.  Air.  Schley  sold  his  cow  for 
$43.20.  He  had  asked  50%  more 
than  the  cost,  and  sold  for  10%  less 
than  the  asking  price.  What  was  his 
gain?     Ans.,  Gain,  $11.20. 

14.  If  cloth  shrinks  121/2%  of  its 
length  in  washing  and  dyeing,  what 
is  gained  in  selling  840  yards  of 
shrunken  cloth  bought  unshrunken  at 
30c  a  yard,  and  sold  at  40c,  the  cost 
of  washing  and  dyeing  being  $4.50? 
Ans.,  $43.50. 

15.  Mr.  Hale  marked  his  suits  at 
20%  above  cost,  and  sold  them  at 
a  discount  of  10%  at  $14.85  each. 
What  was  his  profit  on  250  suits? 
Ans.,  $275. 

16.  Lean  hogs  are  bought  at  8c  a 
pound,  and  fat  rogs  are  sold  at  9c, 
and  it  costs  5c  for  every  pound  a 
hog  increases  in  weight.  Mr.  Cran- 
dall  bought  150  lean  hogs  averaging 
200  pounds  each,  and  increased  their 
weight  25%,  then  sold  them.  How 
much  did  he  make?     Ans.,  $600. 

17.  If  cloth  shrinks  12%  of  its 
length  in  washing  and  dyeing,  how 
many  yards  of  unshrunken  cloth  are 
required  to  make  22  suits,  each  suit 


requiring  12  yards  of  cloth  after  it 
is  washed  and  dyed?  Ans.,  300 
yds. 

18.  Mr.  Hardy  raised  29,280  lbs. 
of  prunes  in  1910.  His  1910  crop 
was  20%  less  than  his  1909  crop, 
and  the  crop  of  1909  was  20%  more 
than  that  of  1908.  How  much  was 
his  1908  crop?     Ans.,  30,500  lbs.? 

19.  Dried  peaches  gain  10%  itl' 
weight  in  processing.  Find  the  prof- 
it on  a  shipment  of  22  carloads  of 
15  tons  each  of  processed  fruit  sold 
at  9c  a  pound ;  the  unprocessed  fruit 
was  bought  at  7c  a  pound,  and  the 
cost  of  packing  and  shipping  was 
$250  per  car.  Commission  at  4% 
was  paid  both  for  buying  and  sell- 
ing.    Ans.,  $7,844. 

20.  A  cubic  centimeter  of  water 
weighs  one  gram.  Gold  is  19.3  and 
silver  10.5  times  as  heavy  as  water. 
What  is  the  weight  of  48  cubic  cen- 
timeters of  an  alloy  of  gold  and  sil- 
ver of  which  75%  is  gold?  Ans,, 
820.8  gr. 

21.  Sea  water  is  2.6%  heavier 
than  fresh  water.  How  much  will 
the  alloy  mentioned  in  problem  20 
weigh  if  suspended  in  sea  water? 
Ans.,  771.552  gr. 

22.  Mr.  Thompson's  fruit  crop  in 
1911  is  80%)  of  his  crop  in  1910,  the 
price  is  50%  higher,  and  the  expense 
of  handling  10%  higher.  He  had  35 
tons  in  1910,  for  which  he  received 
$75  a  ton  and  paid  $5  a  ton  for 
handling.  What  will  his  1911  crop 
net  him?     Ans.,  $2,996. 

23.  An  aviator  traveled  a  certain 
rate  the  first  hour,  increased  his 
speed  20%  the  second  hour,  de- 
creased it  20%  the  third  hour,  and 
increased  it  20%  the  fourth  hour. 
He  traveled  57.6  miles  the  fourth 
hour.  How  far  did  he  travel?  Ans., 
215.6  mi. 

24.  It  is  28  miles  from  San  Jose 
to  Mount  Hamilton.  A  carriage 
travels  75%  faster  coming  down  than 
going  up.  A  carriage  leaves  San 
Jose    at    11  o'clock,    remains    at    the 


PERCENTAGE 


79 


summit  3  hours,  and  returns  at  1 
a.  m.  What  are  the  rates  of  travel 
up  and  back  ?     Ans.,  4  mi. ;  7  mi. 

25.  The  hind  wheel  is  15%  larger 
than  the  fore  wheel  on  a  carriage. 
How  many  revolutions  will  each 
make  while  the  fore  wheel  is  gaining 
45   revolutions?     Ans.,   300;   345. 

26.  Mr.  Barnes  sold  his  farm  for 
$12,340.  He  had  asked  25%  more 
than  the  farm  cost,  and  sold  at  a 
reduction  of  12%  on  the  asking 
price.  What  was  the  gain?  Ans., 
$1,121.82. 

27.  ]\Ir.  W^ells  bought  a  house  and 
lot  for  $4,800.  He  placed  it  on  sale 
at  50%  above  cost,  sold  at  a  re- 
duction of  20%  on  the  asking  price, 
and  paid  5%  of  the  selling  price  to 
the  agent.  What  was  the  gain? 
Ans.,  $672. 

28.  yir.  Wilson  purchased  40  A. 
of  land  at  $75  an  acre,  and  bought 
3,600  orange  trees  at  60c  each.  5% 
of  the  trees  are  found  to  be  worth- 
less, and  20%  of  the  remainder  died. 
He  paid  25c  each  for  having  the 
trees  set  out,  and  $5  an  acre  each 
year  for  cultivation.  What  was  t..-= 
cost  per  living  tree,  including  the 
land,  in  four  years?     Ans.,  $2.43'. 

29.  Mr.  Curtner  sold  his  crop  of 
prunes  in  1912  for  $4,704.  He  had 
40  tons  of  French  prunes.  His  crop 
of  Silver  prunes  was  20%  smaller 
and  the  price  per  ton  20%  higher 
than  the  French  prunes.  At  what 
price  per  ton  did  he  sell  each?  Ans.. 
F.  $60;  S.  $72. 

30.  Mr.  Jackson's  live  stock  is 
valued  at  $3,200.  Forty  per  cent  of 
his  animals  are  goats,  35%  sheep, 
and  the  remainder  hogs.  A  goat  is 
worth  3/i  as  much  as  a  sheep,  and 
a  sheep  is  worth  ^  as  much  as  a 
hog.  Find  the  value  of  each  kind  of 
stock.  Ans..  $900  g. ;  $1,050  sh. ; 
$1,250  h. 

31.  A  hall  committee  paid  $940.80 
for  672  yards  of  carpet.  It  bought 
12  per  cent  more  carpet  than  was 
needed  and  the  price  paid  per  yard 


was  12  per  cent  higher  than  it 
should  have  been.  How  much 
should  the  carpet  have  cost?  Ans. 
$750. 

32.  Green  peaches  lose  85  per 
cent  of  their  weight  in  drying,  and 
dried  peaches  gain  10  per  cent  in 
weight  in  preparation  for  packing. 
It  costs  $5  per  green  ton  for  pick- 
ing and  drying  peaches  and  $2.40 
per  ton  of  packed  fruit  for  packing. 
What  are  the  net  proceeds  of  57^ 
tons  of  packed  fruit  sold  at  7c  a 
pound?     Ans.  $6196.40. 

33.  Twenty  per  cent  of  an  army 
were  killed  in  battle,  30  per  cent  of 
the  remainder  died  of  wounds.  The 
number  which  died  of  wounds  ex- 
ceeded the  number  killed  in  battle, 
472.  How  many  were  left  in  the 
army?     Ans.  6608. 

34.  If  cloth  shrinks  one-ninth  of 
its  length  in  washing  and  dyeing, 
what  is  the  gain  in  selling  720  yards 
of  dyed  cloth,  bought  undyed  at  30 
cts.  a  yard  and  sold  after  being  dyed 
at  40  cts.  a  yard,  if  the  cost  of  wash- 
ing and  dyeing  is  $4.50? 

Ans.,   $40.50. 

35.  The  outer  walls  of  a  building 
contain  5040  square  feet,  and  each 
strip  of  rustic  overlaps  one-eighth  of 
the  width  of  another.  What  will  be 
the  cost  of  the  rustic  and  painting 
for  the  building,  if  the  rustic  costs 
$35  a  thousan  d  and  the  painting 
costs  30  cts,  a  square  vard? 

Ans.,  $369.60. 

36.  If  six  pounds  of  green  coffee 
make  five  pounds  of  dried,  and  green 
coffee  is  bought  at  22  cts  a  pound 
and  roasted  coffee  is  sold  at  30  cts. 
a  pound,  how  much  is  gained  by 
selling  1200  pounds  of  roasted  coffee 
if  the  cost  of  roasting  that  amount 
is  $2.75?  Ans.,  $40.45. 

BUSINESS  CTTSTOMS. 

The  application  of  percentage  to 
different  lines  of  btisiness  should  be 
presented  mainly  from  the  informa- 
tion   standpoint.      The   pupil   will   be 


80 


ARITHMETIC 


interested  in  an  application  only 
when  he  sees  that  it  in  some  way 
touches  the  neighborhood  interests. 
While  he  should  be  encouraged  to 
find  out  for  himself  the  prevailing 
business  customs,  the  teacher  should 
see  that  these  customs  are  fully  and 
clearly  stated,  and  that  they  are 
learned. 

Loss  and  Gain. 

1.  Loss  or  gain  is  reckoned  on 
the  cost. 

2.  Cost  is  100  per  cent  for  reck- 
oning loss  or  gain. 

3.  Selling  price  is  more  than  100 
per  cent  when  there  is   gain. 

3a.  Selling  price  is  less  than  100 
per  cent  when  there  is  loss. 

The  pupil  should  assist  in  mak- 
ing statements  2  and  3. 

Trade  Discount. 

1.  The    First    Discount    is    reck 
oned   on   the   list   price;   the    Second 
Discount    is    reckoned    on    the    first 
proceeds,  and  so  on. 

2.  The  List  Price  is  100  per  cent 
for  reckoning  the  first  discount. 

3.  The  First  Proceeds  is  less 
than  100  per  cent  for  first  discount. 

It  is  100  per  cent  for  reckoning 
the  second  discount,  etc. 

Commission  in   Selling. 

1.  Commission  is  reckoned  on 
the  selling  price. 

2.  The  Selling  price  is  100  per 
cent  for  reckoning  commission. 

3.  The  Proceeds  is  less  than  100 
per  cent. 

Collections  and  similar  transactions 
are  on  the  same  basis  as  selling  on 
commission. 

Commission  in  Buying. 

L  Commission  is  reckoned  on 
the  purchase  price.  (That  paid  by 
the  agent.) 


2,  The  purchase  price  is  100  per 
cent   for   reckoning  commission, 

3.  The  entire  cost  is  more  than 
100  per  cent. 

Property  Insurance. 

When  this  topic  is  taken  up  there 
should  be  a  general  discussion  of  the 
subject,  as  regards  value,  face  of 
policy,  premium,   risk,   etc. 

Fire  insurance  is  usually  quoted 
as  so  many  cents  a  year  on  the  hun- 
dred dollars.  The  three  year  rate 
is  double  the  one  year  rate,  the  five 
year  rate  is  three  times  annual  rate. 
The  agent  usually  receives  15  per 
cent  commission  on  the  amount  of 
premiums  collected,  and  all  the  pol- 
icy fee  when  one  is  charged.  The 
premium  is  reckoned  on  the  face 
value  of  the  policy. 

Life   Insurance. 

Discuss  fraternal  insurance  and 
insurance  companies.  Also  accident 
policies,  endowment  policies,  straight 
life  policies,  who  may  and  who  may 
not  be  insured,  etc. 

The  premium  is  usualy  reckoned 
as  so  many  dollars  per  thousand  on 
the  face  value  of  the  policy,  payable 
annually,  quarterly,  or  monthly  as 
the  case  may  be. 

Taxes  and  Duties. 

Discuss  import  duties,  internal 
revenue,  poll  tax,  property  tax,  their 
purpose,  aim,  and  manner  of  collec- 
tion. 

The  facts  concerning  property 
taxes    in    California   are    as    follows: 

1.  Between  the  first  Monday  in 
March  and  the  last  day  of  June  of 
each  year,  each  property  owner  must 
furnish  the  assessor  with  a  list  ot 
the  property  owned  by  him  at  noon 
on  the   first   Monday   of   March. 

2..  During  the  month  of  July  the 
County  Board  of  Supervisors  sits  as 
a  Board  of  Equalization.  It  ex- 
amines the  valuations  made  by  the 
assessor  and  his  deputies  and  raises 


PERCENTAGE 


81 


or  lowers  any  valuations  as  it  may 
think  proper.  The  property  owner 
has  the  right  to  go  before  the  Board 
and  ask  that  his  assessment  be  low- 
ered, or  show  cause  why  it  should 
not  be  raised. 

3.  In  August  the  State  Board  of 
Equalization  examines  the  assess- 
ments of  the  counties,  and  raises  or 
lowers  any  county  assessment  as  it 
may  think  proper.  Each  county  has 
a  right  to  be  heard  thru  its  super- 
visors. 

The  State  Board  also  places  a 
valuation  on  the  railway  property  of 
the  State,  and  apportions  this  valu- 
ation among  the  counties  in  propor- 
tion  to  the  number  of  miles  of  track 
in  each  county. 

It  also  fixes  the  State  tax  rate. 

4.  In  September  the  Board  of 
Supervisors  fixes  the  county  and 
city  tax  rates. 

5.  The  Auditor  calculates  the  tax 
of  each  individual  and  must  have  his 
work  completed  by  the  second  Mon- 
day in  October. 

6.  Taxes  are  payable  to  the  Tax 
Collector  in  two  installments.  The 
first  installment  consists  of  all  the 
tax  on  personal  property,  and  half 
the  tax  on  real  estate.  It  is  delin- 
quent if  not  paid  on  or  before  the 
last  Monday  in  November,  and  a 
penalty  of  15  per  cent  is  added. 

The  second  installment  is  half  the 
tax  on  real  estate  and  is  delinquent 
if  not  paid  on  or  before  the  last 
Monday  in  April  and  5  per  cent  is 
added  to  all  taxes  remaining  unpaid. 
If  not  paid  before  the  delinquent 
tax  list  is  published  a  charge  of  50 
cents  is  added  for  each  piece  of 
property  delinquent.  If  not  paid  the 
property  is  sold  to  the  State. 

The  teacher  should  show  the  pu- 
pils an  assessment  blank  and  a  tax 
receipt,  also  a  delinquent  tax  list. 

In  solving  problems  in  loss  and 
gain,  commission,  etc.,  the  work 
should  be  written  in  good  business 
form.  Business  blanks,  bill  heads, 
and  ruled  journal  and  ledger  paper 
should    be     used     when    practicable. 


The  required  multiplications  and  di- 
visions should  be  performed  as  side 
work,  and  in  the  additions  and  sub- 
tractions the  decimal  points  should 
be  kept  in  the  same  vertical  column, 
and  when  there  are  several  items  oi 
debits  and  credits  double  columns 
should  be  used. 

A  few  models  are  here  given. 

1.  Mr.  Copeland  marked  his 
suits  40  per  cent  above  cost  and  sold 
them  at  10  per  cent  discount  on  the 
marked  price,  receiving  $24.57  each. 
How  much  was  his  profit  on  50 
suits  ? 

Let  100  per  cent  equal  cost  of  one 
suit. 

Then  140  per  cent  equals  marked 
price. 

14  per  cent  equals  reduction, 

126  per  cent  equals  selling  price. 

126  per  cent  of  cost  equals  $24.57 

cost  of  one  suit  equals    .  . .  .$19.50 


gain    on   suit   equals    . . . 

,..$  5.07 

gain  on  50  suits  equals.  . 

.  .$253.50 

Side  work. 

$19,50 

1.26)24.57' 00 
12.6 


1197 
1134 


630 

630 

$5.07 
50 

$253.50 

2.  Miss  Wise  bought  a  lot  for 
$840  and  built  a  house  costing 
$2800.  She  rents  the  house  at  $35 
a  month  and  pays  $1.60  a  month  for 
water,  $55  a  year  for  taxes  and  in- 
surance on  ^  of  the  cost  of  the 
house  at  an  annual  rate  of  45c. 
What  per  cent  does  her  investment 
net  her? 

Solution. 

Receipts, 

Rent  for  12  months  .... 


.$420 


82 


ARITHMETIC 


Expenses. 
Water  for  12  months  $19.20 

Taxes    55.00 

Insurance   9 .  45     83 .  65 

Net  receipts    $336 .  35 

Cost  of  property    $3640 

Rate   of  interest    9.2  per   cent 

Side  work. 

.092 


3640)336.350 
327  6 


8  75 

7  28 


1  47 

3.  Mr.  Harmon  is  assessed  $12400 
on  real  estate  and  $950  on  personal 
property.  The  tax  rate  is  $1.76.  He 
pays  all  his  taxes  May  2nd.  How 
much  does  he  pay  ? 

Yz   real  estate  tax    $109.12 

Personal   property    16 .  72 

First    Installment     125 .  84 

Second    Installment     109.12 

Penalty   first   installment    .  .     25 .  168 
Penalty   second   installment.       5.456 

Amt.    paid $265.58 

4.  A  commission  merchant  sold 
on  commission  of  5  per  cent  500 
sacks,  55,000  lbs.  of  potatoes  @ 
$1.20  per  C;  600  melons  @  $9.25 
per  C.  He  paid  $45  freight  and 
$8.50  drayage.  Find  the  amount 
due  the  consignor. 

55000    lb.    potatoes 

@  1.20  per  C... $660. 00 
600  melons 

@  9.25  per  C...     55.50    $715.50 

Commission  at  5%     35.78 

Freight 45 . .  . 

Drayage 8.50         89.28 

Balance $626.22 

Problems. 

1.  Mr.  Barnes  sold  his  farm  for 
$12,340.  He  had  asked  25%  more 
than   the    farm   cost,    and    sold    at   a 


reduction  of  12%  on  the  asking" 
price.  What  was  the  gain?  \ni,., 
$1,121.82. 

2.  Mr.  \\'ells  bought  a  house  ana 
lot  for  $4,8U0.  He  placed  it  on  sale 
at  50%  above  cost,  sold  it  at  a  re- 
duction of  20%  on  the  asking  price, 
and  paid  5%  of  the  selling  price  to 
the  agent.  \\'hat  was  the  gain? 
Ans.,  $672. 

3.  Mr.  Johnson  bought  576  sacks 
of  potatoes  at  ,$1.20  a  sack.  12>4% 
of  them  spoiled.  At  what  price  per 
sack  must  he  sell  the  remainder  to 
reahze  a  profit  of  16%%  on  the 
whole  investment?     Ans.,   $1.60. 

4.  Mr.  Johnson  bought  576  sacks 
of  potatoes  at  $1.20  per  sack.  16%% 
of  them  had  to  be  sold  at  a  loss  of 
25%.  At  what  price  per  sack  must 
the  remainder  be  sold  to  realize  a 
profit  of  20%  on  the  whole  invest- 
ment?    Ans.,  $1,548. 

5.  Mr.  Conkling  bought  500  box- 
es of  oranges  at  $1.20  per  box. 
40%  being  large,  were  sold  at  a 
profit  of  12j/2%.  At  how  much  per 
box  must  the  rest  be  sold  to  realize 
a  profit  of  25%  on  the  whole  in- 
vestment?    Ans.,   $1.60. 

6.  Mr.  Stockton  and  Mr.  Thomp- 
son each  agreed  to  sell  7,038  sacks 
of  grain.  It  was  found  that  Mr. 
Stockton  had  underestimated  his  crop 
20%  and  ]Mr.  Thompson  had  over- 
estimated his  crop  20%.  How  much 
has  each?  Ans.,  S.  8,797.5:  T. 
5,865. 

7.  Air.  A  purchased  a  mower  Hst- 
at  $160,  at  30  and  15  off.  He  paid 
an  agent  5%  for  making  the  pur- 
chase, and  freight  of  $6.25.  He  sold 
the  machine  at  10%;  off  the  list  price. 
What  was  the  gain?    Ans.,  $37.79. 

8.  Mr.  B  sold  a  house  for  $7,500, 
which  was  25%  more  than  it  cost. 
He  paid  an  agent  5%  for  making 
the  sale.  With  the  proceeds  he 
bought  another,  which  was  after- 
ward sold  at  a  loss  of  10%,  no  com- 
mission. What  was  the  net  loss  or 
gain?     Ans.,  $412.50. 


PERCENTAGE 


83 


9.  Mr.  Hale  bought  a  house  for 
$4,200,  and  spent  $300  for  repairs. 
He  offered  it  for  sale  at  $7,000,  and 
aferward  sold  it  at  a  reduction  of 
10%,  and  paid  his  agent  5%  com- 
mission. How  much  did  Mr.  Hale 
make?     Ans.,  $1,485. 

10.  Mrs.  Prim  bought  a  house 
for  $3,600  and  spent  $200  in  altera- 
tions and  repairs.  She  pays  a  yeav- 
ly  tax  at  $2.50  on  an  assessed  valu- 
ation of  $2,250,  insurance  on  the 
three  years'  plan  on  $2,500  at  the 
annual  rate  of  $.45,  and  a  m.onthly 
water  rate  of  $1.80.  If  the  house 
is  kept  rented  at  $35  a  month,  what 
per  cent  does  the  investment  net 
her?     Ans.,  8.8%. 

11.  Mr.  Clark  bought  a  store  at 
$8,000  which  he  rents  at  $100  a 
month.  He  pays  insurance  at  $1.10 
on  $6,000,  taxes  at  $2.55  on  $4,800, 
and  estimates  a  yearly  repair  bill  of 
$100.  What  per  cent  does  the  in- 
vestment net  him?     Ans.,  11.4%. 

12.  Mr.  Martin  bought  a  horse 
for  $125,  offered  it  for  sale  at  40% 
above  cost,  and  sold  it  at  a  reduction 
of  10%.  He  paid  an  agent  5%  for 
making  the  sale,  and  $12.50  for  oth- 
er expenses.  How  much  did  he 
gain?     Ans.,  $12.13. 

13.  Mrs.  Dawson  offered  her 
house  for  sale  at  25%  above  cost, 
and  afterward  sold  it  at  a  reduction 
of  12%.  The  agent  got  5%  com- 
mission, and  she  received  $4,723.40. 
What  did  she  m.ake  or  lose?  Ans., 
$203.40  gain. 

14.  Mr,  Buell  bought  a  carriage 
listed  at  $500,  receiving  discounts  of 
25  and  10  off  and  paying  an  agent 
5%  for  making  the  purchase  and 
$12.50  freight.  He  sold  the  carriage 
at  15%  discount  on  the  list  price, 
paying  an  agent  4%.  What  was  his 
gain?     Ans.,  $41.12. 

15.  A  house  was  kept  insured  on 
the  three  years'  plan  for  $2,500  at 
an  annual  rate  of  45c.  It  burned 
the  eleventh  year.  Find  the  cost  of 
insurance,   including  a  policy   fee   of 


$1    for   each   issuance   of   his   policy. 
Ans.,  $94. 

16.  How  much  did  the  agent 
make  and  how  much  did  the  com- 
pany receive  from  the  transactions 
in  problem,  fifteen  ?  Ans.,  A.  $17.50 ; 
Co.  $76.50. 

17.  Mr.  Kirk  bought  a  lot  for 
$800  and  built  a  house  costing 
$2,400,  insures  his  house  for  $2,000 
at  50c,  pays  taxes  at  $2.60  on  an 
assessed  valuation  of  $1,920.  For 
how  much  per  month  must  he  rent 
the  house  to  realize  8%  net  on  bis 
money?     Ans.,  $26.33. 

18.  Mr.  A  bought  a  piano  listed 
at  $800  at  40  and  20  off.  He  paid 
an  agent  10%  for  purchasing,  and 
paid  $45  for  freight.  He  sold  it  at 
15%  discount.  What  was  tlie  gain? 
Ans.,  $212.60. 

19.  Mr.  Christie  bought  a  piano 
listed  at  $700  at  25  and  15  off.  He 
afterward  sold  it  to  Mrs.  Monroe, 
receiving-  $50  cash  and  a  monthly 
payment  of  $6.  After  making  seven 
payments  Mrs.  Monroe  returned  the 
instrument.  Mr.  Christie  spent  $5 
for  repairs  and  then  sold  the  piano 
for  $550.  What  was  his  total  profit? 
Ans.,  $190.75. 

20.  Mr.  Sherman  bought,  through 
an  agent,  a  carload,  20  tons,  of 
wheat  at  $1.75  per  cwt.,  and  paid 
$15  freight.  Pie  sold  the  same  at 
$1.95  per  cwt.  He  paid  4%  com- 
mission for  buying,  and  2%%  for 
selling.  What  was  his  gain?  An?.. 
$17.50. 

21.  A  commission  merchant 
bought  for  Mr.  ]\Iadsen  20  doz. 
chairs  listed  $45  per  doz.,  at  dis- 
counts of  20  and  10,  and  sold  the 
same  at  15%  discount,  com.  for  buy- 
ing 2%,  com.  for  selling  4%,  other 
expenses  $8.75.  What  was  Mr. 
Madsen's   profit?     Ans.,   $64.69. 

22.  Wiley  B.  Allen  bought  a 
piano  listed  at  $600  at  30  ana  10 
off,  and  paid  an  agent  5%  for  mak- 
ing the  purchase.  He  sold  the  same 
piano  at  20%   discount  and  paid  his 


84 


ARITHMETIC 


clerk  4%  for  making  the  sale.    How 
much   did   he   make?     Ans.,    $63.90. 

23.  Mrs.  Jamison  bought  a  house 
and  lot  for  $3,000  and  spent  $500 
for  alterations  and  repairs.  She  in- 
sured the  house  for  $2,800  on  the 
three  years'  plan  at  a  basis  rate  of 
45c,  and  pays  taxes  at  $2.45  on  an 
assessed  valuation  of  $2,250.  The 
house  rents  at  $40  per  month,  and 
the  water  rate  is  $2  a  month.  What 
per  cent  does  the  investment  pay? 
Ans.,  11.21%. 

24.  Mr.  Strong  keeps  his  house 
insured  on  the  three  years'  plan  for 
$3,560,  the  annual  rate  being  65c, 
poHcy  fee  $1.  The  house  burns 
during  the  thirteenth  year.  What 
does  the  agent  make,  and  how  much 
does  the  company  receive?  Ans., 
A.  $39.71;  Co.  $196.69. 

25.  Mrs.  Phillips  bought  a  house 
and  lot  for  $2,400,  keeps  it  insured 
on  the  three  years'  plan  for  $1,600, 
at  the  annual  rate  of  45c,  no  policy 
fee.  She  pays  taxes  at  $2.45  on 
$1,500,  water'  at  $1.60  per  month, 
and  rents  the  house  for  $25  a  month. 
The  house  is  vacant  two  months 
each  year.  What  per  cent  does  she 
realize  on  her  money?  (No  water 
tax  when  the  house  is  vacant.) 
Ans.,  83/io„%. 

26.  Mr.  Mason  bought  a  house 
and  lot  for  $3,800  and  made  altera- 
tions costing  $700.  He  placed  it  on 
sale  for  $6,000,  afterward  reduced 
the  price  10%  and  paid  the  agent 
5%.  What  per  cent  does  he  make 
on  his  investment?     Ans.,  14%. 

27.  The  Home  Union  bought 
flour  listed  at  $4.75  a  bbl.  at  10% 
discount,  paid  freight  at  25c  per 
bbl.,  and  sold  the  consignment  at 
5%  above  list  price.  How  much 
was  made  on  1,200  bbls.  ?  Ans., 
$555. 

28.  Miss  Thrifty  bought  a  lot  for 
$1,200,  built  a  house  for  $3,200,  in- 
sured it  for  three  years  on  %  of 
value  at  40c  annual  rate.  She  rent- 
ed it  at  $35  a  month,  reservcvl  $25 
a  year   for  repairs.     She  paid  taxes 


at  $2.55  regular  and  35c  special,  on 
a  valuation  of  $2,500,  and  pays  wa- 
ter rate  of  $1.50  a  month.  Find  per 
cent  on  investment.     Ans.,  6.775%. 

29.  Mr.  Ross  bought  a  house  for 
$3,000  and  spent  $200  for  repairs. 
He  offered  it  for  sale  at  $4,000,  re- 
duced his  price  10%,  and  paid  an 
agent  5%  for  selling  it.  What  was 
his  gain?     Ans.,  $220. 

30.  Mr.  Lion  sells  furniture  on 
credit  at  an  advancement  of  25%  on 
the  cost.  He  collects  through  an 
agent  90%  of  the  sales.  The  age*t 
keeps  4%  commission,  and  pays  in 
$43,200.  What  is  Mr.  Lion's  profit, 
and  what  per  cent  does  he  make  on 
his  goods?     Ans.,  $3,200;  8%. 

31.  A  contractor  builds  a  House 
for  $3,300,  realizing  a  profit  of  20% 
on  the  cost.  The  cost  of  the  labor 
was  to  the  cost  of  the  material  as  2 
is  to  3.  Find  cost  of  labor  and  ma- 
terial.    Ans.,  L.  $1,100;  M.  $1,650. 

32.  If  wages  should  advance 
20%,  and  material  decline  20%  in 
value,  and  the  same  house  be  built 
at  the  same  price,  wh^t  per  cent 
would  the  contractor  make  on  the 
cost?     Ans.,  25%. 

33.  Mr.  Harper's  property  is  as- 
sessed as  follows :  real  estate,  $6,500, 
personal  property,  $1,500.  The  reg- 
ular rate  is  $2.45,  the  special  rate 
25c.  Find  each  installment  of  his 
taxes.     Ans.,  $128.25,  $87.75. 

34.  How  much  will  Mr.  Harper's 
taxes  be  if  they  are  all  paid  January 
10th?     Ans.,  $235.24. 

35.  Mrs.  Hooker  is  assessed 
$4,500  on  real  estate  and  $500  on 
personal  property.  The  rates  are 
$2.45  regular  and  25c  special.  She 
pays  her  taxes  May  1st.  How  much 
does  she  pay?     Ans.,  $152.89. 

36.  Mr.  Bennett's  house  was  as- 
sessed at  $2,400.  and  his  personal 
property  for  $200.  The  regular  rate 
was  $2.10,  and  the  special  rate  15c. 
He  paid  his  first  installment  Febru- 
ary  1st,    and   his    second    installment 


INTEREST 


85 


May  2nd.     What  did  his  taxes  cost 
him?     Ans.,  $64.58. 

37.  Mr.  Foley  is  assessed  $12,400 
on  real  estate  and  $2,600  on  per- 
sonal property.  The  regular  rate  is 
$1.55,  and  the  special  rate  15c.  Mr. 
F.  pays  the  first  installment  April 
1st,  and  the  second  installment  May 
1st.  How  much  do  his  taxes  cost 
him?     Ans.,  $282.71. 

38.  Mrs.  Anderson  has  real  es- 
tate assessed  at  $6,400  and  personal 
property  assessed  at  $3,640.  The 
regular  state  and  county  rate  is 
$1.65,  and  there  is  a  special  rate  6i 
15c.  Find  each  installment  of  her 
taxes.     Ans.,  $123.12 ;  $57.60. 

39.  What  will  her  taxes  be  if  paid 
March   1st?     Ans.,   $199.19. 

40.  What  will  her  taxes  cost  if 
all  are  paid  May  1st?    Ans.,  $208.22. 

41.  Mr.  Reynolds  sold  a  house  at 
a  loss  of  25  per  cent.  He  invested 
the  money  received  in  another  house 
which  he  afterward  sold  for  $4,104, 
gaining  20  per  cent  on  its  cost.  What 
was  his  net  loss.     Ans.,  $456. 

42.  Mr.  Conkling  sold  40  per  cent 
of  a  carload  of  potatoes  at  a  profit 
of  50  per  cent,  25  per  cent  at  a 
profit  of  20  per  cent  and  the  re- 
mainder at  a  loss  of  SSy^  per  cent. 
He  received  altogether  $408.  W^hat 
was  his  gain?     Ans.,  $48. 

43.  Mr.  Anderson  bought  a  piano 
listed  at  $600,  with  discounts  of  40 
and  20  off.  He  sold  it  at  a  discount 
of  25  per  cent  and  paid  an  agent  10 
per  cent  commission  for  selling.  What 
was  Mr.  Anderson's  gain,  and  what 
was   the   agent's   commission? 

Ans.,  $117. 

44.  Mr.  Buell  bought  a  carriage 
listed  at  $500,  receiving  discounts  of 
25  and  10  off,  paying  an  agent  5 
per  cent  for  making  the  purchase, 
and  $12.50  freight.  He  sold  the  car- 
riage at  15  per  cent  discount.  What 
was  his  gain?  Ans.,  $59,125. 

45.  Mr.  Cowper  received  $114  as 
the  proceeds  of  the  sale  of  a  mower. 


He  had  allowed  a  discount  of  4  per 
cent  and  had  paid  an  agent  5  per 
cent  commission  for  making  the  sale. 
The  mower  had  been  purchased  at 
30  per  cent  discount.  What  was 
Mr.   Cowper's  gain?       Ans.,  $26.50. 

INTEREST. 

Interest  is  usually  charged  at  a 
certain  rate  per  cent  per  annum. 

The  time  is  found  by  counting 
from  one  date  to  another,  ordinarily 
by  compound  subtraction,  sometimes 
however,  by  finding  the  actual  number 
of  days.  When  compound  subtraction 
is  used,  30  days  are  called  a  month 
and  12  months,  or  360  days,  a  year. 
When  the  time  is  found  by  counting 
the  actual  number  of  days  360  days 
are  called  a  year  in  ordinary  com- 
mercial transactions,  and  for  exact 
interest  365  days  make  a  year.  Ex- 
act interest  is  reckoned  by  the  large 
city  banks  only,  and  by  the  United 
States    government. 

When  interest  is  to  be  calculated 
for  years  or  years  and  months,  it  is 
only  necessary  to  find  the  interest 
for  one  year  and  then  multiply  this 
by  the  number  of  years  and  frac- 
tion of  a  year. 

Find  the  interest  on  $265.40  for 
3  years  9  months  at  5  per  cent. 

Solution. 
3   yr.   9   mo.   equal   3^   years. 
$264.40 
.05 


$13.2200  Int.  for  1  yr. 

m 


9915 
3966 


$49.58  Int.  3^  yr. 

When  interest  is  to  be  found  for 
days  or  months  and  days  any  one 
of  several  different  methods  may  be 
used. 

Cancellation  Method. 

Reduce  the  time  to  a   fraction  of 


86 


ARITHMETIC 


a    year,    indicate   the    work    and    use 
cancellation. 

Find    the   interest   on   $375.60    lor 
7  months  18  days  at  5  per  cent. 
Solution. 
7  mo.  18  da.  equal  ^%o  yr. 
.1252 

0  X  5  X  19 


10P      ^p 

$.1252X5X19=$11.894  Ans. 

To  apply  the  method  skillfully  re- 
quires that  attention  be  given  to 
three  things: — 

1st — Reducing  the  time  to  a  frac- 
tion of  a  year. 

2nd — Indicating  the  work. 

3rd — The  canceling. 

First.  It  is  best  as  a  rule  to 
change  the  days  to  a  fraction  of  a 
month,  unite  the  result  with  the 
months,  then  change  to  a  fraction  of 
a  year.  When  days  only  are  given, 
write  360ths  of  a  year  and  reduce 
to  its  lowest  terms.    Thus, 

6  mo.  18  da.  =  6%  mo.  =  i%o  y- 
4  mo.  8  da.  =  4%5  mo,  =  i%5  yr. 

7  mo.  20  da.  =7^  mo.  =  2%g  yr. 
4  m.o,  15  da.  =4%  mo.=^  yr. 
2  yr.  4  mo.  15  da.  =  2^$  yr. 

Reduce  to  fractions  of  a  year  3 
mo.  18  da.,  5  mo.  12  da.,  2  yr.  4  mo. 
9  da.,  1  yr.   7  mo.    10  da. 

It  should  be  noted  that  the  factors 
of  30  are  2-3-5;  those  of  12,  2-2-3. 
If  the  number  of  days  does  not  con- 
tain a  factor  2,  3,  or  5,  it  is  best 
to  reduce  the  months  and  days  to 
days  and  place  the  result  over  360. 

When  exact  interest  is  required 
find  the  actual  number  of  days  and 
place  over  365. 

Second — Write  the  rate  as  a  com- 
mon fraction  and  express  the  time 
as  a  proper  or  improper  fraction  as 
the  case  may  require.  See  examples 
solved  above. 

Third — It  is  best,  as  a  rule,  to  can- 
cel the  ciphers  in  the  denominators 
first  and  place  a  mark  in  the  dollars 
as  many  places  to  the  left  of  the  dec- 


imal point  as  there  are  ciphers  in  the 
denominators  which  have  been  can- 
celed. Do  not  cancel  a  cipher  to 
the  right  of  the  decimal  point,  for 
this  makes  no  change  in  the  value  of 
the  number.  When  other  factors 
have  been  canceled  into  the  dollars, 
the  mark  should  be  replaced  by  the 
decimal  point. 

Wh*^  the  denominator  has  been 
redif  ,a  to  a  number  not  greater 
than  12  so  that  short  division  may 
be  used,  little  is  gained  by  further 
cancellation.  It  is  as  easy  to  divide 
by  9  as  by  3,  by  8  as  by  4. 

Six  Per  Cent  Method. 

First  get  the  interest  on  $1  for  the 
given  time  at  6  per  cent. 

The  interest  on  $1  for  1  vear  is 
$.06. 

The  interest  on  $1  for  2  montus 
is  .01. 

The  interest  on  $1  for  1  month  is 
.005. 

The  interest  on  $1  for  6  days  is 
.001. 

The  interest  on  $1  for  1  dav  is 
.000^. 

Hence   the    rule : — 

To  get  the  interest  on  $1  for  any 
time  at  6  per  cent.  Multiply  .06  by 
the  number  of  years.  Divide  the  num- 
ber of  months  by  2,  call  the  quotient 
cents  and  the  remainder  if  any  5 
mills.  Divide  the  number  of  days  by 
6,  call  the  quotient  mills  and  the  re- 
mainder Bths  of  a  mill.  The  sum  of 
the  results  is  the  required  interest. 

Find  the  interest  on  $1   for  3  yr. 
7  mo.  19  da.  at  6  per  cent. 
$1   @  6  per  ct.   for  3  yr  =  .18 
$1  @  6  per  ct.  for  7  mo.  =  .035 
$1  @  6  per  ct.  for  19  da.  =  .003^^6 


Total  =  .218>^ 

With  a  little  practice  the  result 
may  be  found  by  inspection.  Fol- 
low the  order  given  above,  and  add 
results  as  you  proceed.  Thus:  .18, 
.215,   .218^. 


COAIPOUND   INTEREST 


87 


For  4  yr.  9  mo.  24  da.  The  re- 
sults are  thought  out  as  follows : 

.24,   .285,   .289. 

Find  the^interest  on  $1  at  6  per 
cent  for 

2  yr.  8  mo.  18  da. 

5  yr.  6  mo.  12  da. 

7  yr.  7  mo.  15  da. 

1  yr.  9  mo.  11  da.  etc. 

Second.  To  find  the  interest  on 
any  principal  for  any  time  at  6  per 
cent.  Multiply  the  principal  by  the 
interest  on  $1  at  6  per  cent  for  the 
given  time. 

Find  the  interest  on  $247.40  for  3 
yr.  10  mo.  18  da. 

Interest  on  $1  for  the  given  time 
is  .233. 

$247.40     Prin. 

.233     Int.  on  $1. 


74220 
74220 
49480 


$57.64420  Ans. 

Third.  To  find  the  interest  on  any 
principal  at  any  rate.  Find  the  in- 
terest at  6  per  cent  and  increase  or 
decrease  the  result  by  such  a  fraction 
of  itself  as  the  per  cent  is  greater  or 
less  than  6. 

Use  of  Interest  Tables. 

Books  of  tables  are  prepared  giv- 
ing the  time  from  any  date  to  any 
other  in  the  year,  and  data  from 
which  the  interest  on  any  sum  at 
any  ordinary  rate  for  any  desired 
number  of  years,  months  and  days 
may  be  obtained  by  addition.  These 
books  are  used  largely  by  bankers 
and  others  who  have  much  interest 
calculating  to  do. 

Solve  the  first  four  problems  giv- 
en below  by  cancellation,  the  next 
four  by  the  six  per  cent  method,  the 
next  by  either  method,  and  the  last 
two  by  using  the  actual  number  of 
days  and  365  days  to  the  year. 


Find  the  interest: 

1.  On  $485.40  at  7  per  cent  from 
Jan.   10  to  July  25,   1914. 

2.  On  $296.52  at  4y2  per  cent 
from  Dec.  16,  1913,  to  Aug.  4, 
1914. 

3.  On  $956  at  8^  per  cent  from 
Sept.  10,  1914,  to  Jan.  26,  1915. 

4.  On  $76.85  at  9  per  cent  from 
Feb.  13,  1915,  to  May  17,  1916. 

5.  On  $283.56  at  6  per  cent  from 
Oct.  8,  1914,  to  April  26,  1916. 

6.  On  $4967.25  at  6  per  cent  from 
May  7,  1913,  to  Jan.  16,  1916. 

7.  On  $274.45  at  5  per  cent  from 
Nov.  6,  1913,  to  June  1,  1914. 

8.  On  $865.27  at  7^2  per  cent 
from  Oct.  18,  1914,  to  Feb.  28,  1916. 

9.  On  $1360.32  at  5%  per  cent 
from  May  21,  to  Dec.  14,  1915. 

10.  On  $675.80  at  8  per  ceni 
from  April  17,  1914,  to .  July  24, 
1915. 

11.  On  $674.82  at  6  per  cent  for 
3  yr.  7  mo.  11  da. 

12.  On  $206.15  at  4  per  cent  for 
7  yr.  3  mo.  18  da. 

13.  On  $267.83  at  6  per  cent 
from  June  27  to  Sept.  23,   1914. 

14.  On  $7658.25  at  5  per  cent 
from  Feb.  18  to  May  27,  1916. 

COMPOUND  INTEREST. 

When  interest  is  made  payable  at 
stated  intervals,  if  it  is  not  paid 
when  due,  it  is  usually  added  to 
the  principal  and  bears  interest.  In 
such  cases  interest  is  said  to  be 
compounded.  Postal  and  other  sav- 
ings banks  pay  compound  interest. 

When  interest  is  to  be  compound- 
ed the  result  for  long  terms  is  found 
by  the  use  of  compound  interest  t4- 
bles. 

When  the  use  of  a  table  is  not 
convenient,  the  interest  is  added  to 
the  principal  at  the  end  of  each  term, 
and  this  becomes  the  principal  for 
the  succeeding  term  or  part  of  a 
term  as  the  case  may  be. 


88 


ARITHMETIC 


A  good  concise  form  saves  timt 
and  is  a  safeguard  against  mistakes. 

Find  the  compound  interest  on 
$375  for  1  yr.  9  mo.  10  da.  at  4  per 
cent  compounded  semiannually. 

Solution. 

4  per  cent  is  2  per  cent  for  a  half 
year  or  term. 

1  yr.  9  mo.  10  da.  =  3  terms  +  3 
mo.  10  da. 

3  mo.  10  da.  =  %  of  a  term 
1st  Prin.  $375.  X  .02 
1st  Int.  7.50 


2nd  Prin         382.50  X 
2nd  Int.               7.65 

.02 

3rd  Prin.         390.15  X 
3rd  Int.               7.803 

.02 

4th  Prin.         397. 953 X 
4th  Int.               4.422 

.02    X  % 

Amt.          $402,375 
Prin.          $375. 

Int.             $  27.375 
Side  work: 
39'7.953  X2X^— 39.7953— 4.422 

100    9              9 
10 

1.  Find  the  amount  of  $2100  for 
2  yr.  6  mo.  at  6  per  cent  compound- 
ed  semiannually.     Ans.  $2434.48. 

2.  A  note  for  $1200  dated  Nov. 
7,  1912,  with  interest  at  8  per  ceni 
compounded  semi-annually  was  paid 
Jan.  1st,  1915.  How  much  was  due, 
no  payment  having  been  made,  Ans., 
$1420.69. 

3.  Henry  Moss  deposits  $100  in  a 
savings  bank  Jan.  1  and  July  1  of 
each  year  beginning  Jan.  1, 1914.  The 
bank  allows  4  per  cent  interest  com- 
pounded semi-annually.  If  the  prac- 
tise is  continued  and  no  money  is 
withdrawn  how  much  will  be  due 
him  Dec.  31.  1916?    Ans.,  $643.43. 

4.  Mr.  Allen  has  a  school  bond 
for  $1000  bearing  6  per  cent  interest 
payable  semi-annually,  the  bond  to 
be  paid  in  five  years.     If  Mr.  Allen 


deposits  his  interest  payments  in  a 
bank  which  pays  interest  at  4  per 
cent  payable  semi-annually,  how 
much  will  he  have  to  his  credit  at 
the  end  of  the  five  years  inclusive 
of  the  final  payment?  Ans.,  $1328.47. 

5.  Miss  Stoner  opened  an  ac- 
count with  a  savings  bank  Jan.  1, 
1914,  and  deposits  $5  each  Thursday. 
The  bank  pays  4  per  cent  compound 
interest  allowing  interest  on  the 
amount  on  deposit  on  the  first  day 
of  any  month  and  adds  the  interest 
July  1st  and  Jan.  1st.  What  will 
be  due  on  her  account  Jan,  1,  1915? 

Ans.,  $269.89. 

6.  Find  the  amount  of  $376  for 
1  yr.  8  mo.  10  da.  at  6  per  cent 
compounded  quarterly.  Ans.,  $415.93. 

Partial  Payments. 

It  is  not  unusual  for  partial  pay- 
ments to  be  made  on  notes  before 
the  time  for  final  settlement.  The 
study  of  Partial  Payments  is  there- 
fore of  value  in  itself,  besides  giv- 
ing practice  in  computing  interest, 
in  the  use  of  good  forms  and  in 
keeping  track  of  the  work  in  a  series 
of  operations. 

Before  commencing  the  work  the 
data  should  all  be  taken  down  as 
shown  below,  each  date  and  its  cor- 
responding payment  being  written 
above  the  date  and  payment  immedi- 
ately preceding  it. 

Next,  all  the  times  should  be 
found  in  order  by  compound  sub- 
traction and  the  corresponding  pay- 
ments  brought  down.  Then  the  cal- 
culations should  proceed  step  by  step, 
care  being  taken  that  the  interest 
calculations  are  clearly  indicated, 
that  in  the  addition  and  subtraction 
work  the  decimal  points  are  kept  in 
the  same  vertical  line,  and  that  each 
payment  is  checked  as  soon  as  it  is 
used. 

The  United  States  rule  is  com- 
monly followed  when  the  note  runs 
more  than  a  year.  When  the  time 
is  a  year  or  less  the  Mercantile  rule 
is  usually  followed,  tho  there  are  no 
fixed  customs  for  either  rule. 


PARTIAL  PAYMENTS 


89 


United  States  Rule.  Find  the 
amount  of  the  principal  to  a  time 
when  a  payment  or  the  sum  of  two 
or  more   payments   equals   or   exceeds 


the  interest  due,  and  from  the  amount 
subtract  such  payment  or  paj^ents. 

With  the  remainder  as  a  new  prin- 
cipal proceed  as  before. 


Model  I. 

(Each  payment  exceeds  the  interest 

due.) 


A  note  for  $5000  was  given  Aug. 
2,  1908,  bearing  interest  at  6  per 
cent.  The  following  payments  were 
endorsed:  Paid  Sept.  9,  1908,  $500; 
May  12,  1909,  $350.  What  amount 
would  settle  the  note  July  2,   1909? 


Yr. 
1909 
1909 
1908 
1908 


Mo. 
7 
5 
9 
8 


Da. 

—  2 

—  12 

—  9 

—  2 


$350 
$500 


.0061/6—     1     —     7     $500  V 
.0405  —8    —    3     $350V 
.00%  —    1    —  20    ^.^^V 
5000      X   .006>^=$  30.833 
4530. 83X   .0403   =$183,498 
4364. 33X    .00%    =$  36.369 


Rate  6% 


Prin. 
Int. 

Amt. 
1st  Pay. 

Bal. 
Int. 

Amt. 
2nd  Pay. 

Bal. 
Int. 

Amt. 

500 
30 

83 

5030 
500 

83 

4530 
183 

83 
50 

4714 
350 

33 

4364 
36 

33 
37 

4400 

70 

Model  II. 
(The  interest  exceeds  a  payment.) 
Principal  $850.  Date  May  10, 
1905.  Rate  7  per  cent.  Endorse- 
ments:—July  15,  1906,  $130;  June 
1,  1907,  $46;  Dec.  12,  1908,  $380. 
What  was  due  May  10,  1909? 


Yr.     Mo.  Da. 

1909—  5—10  

1908—12—12  $380 

1907—  6—1  $  46 

1906—  7—15  $130 

1905—  5—10  

1_  2—  5  $130V 

—10—16  $  46V 

1—  6—11  $380  V 

-28  


8'50X7><85=$70.24 

Z0P    72 
'790 .  24  X7X79=$48 .  556 

;0P    90 
790 .  24  X  7X551=$84 .657 

;00  260 
'497 .  46  X7X37:^$14 .  315 

m   90 


Rate  7% 


Prin. 
Int. 

Amt. 
1st   Pay. 

Bal. 

■Int. 

Int. 

Amt. 

2nd  &  3rd  Pay. 

Bal. 
Int. 

Amt. 


850 


70 

24 

920 
130 

24 

790 
48 
84 

24 
56 
66 

923 

426 

46 

497 
14 

46 
32 

511 


78 


90 


ARITHMETIC 


1.  A  note  for  $2150,  dated  Mar. 
10,  1913,  and  bearing  interest  at  8 
per  cent  was  indorsed  as  follows: 

Sept  25,  1913,  $275. 
Mar.    10,  1914,  $365. 
How    much    was    due    Mar.     10, 
1915?     Ans.,   $1770.39. 

2.  A  note  for  $965,  dated  Jan. 
16,  1912,  and  bearing  interest  at  7 
per  cent  has  the  following  indorse- 
ments : 

May  7,  1913,  $150. 
Jan.   16,  1914,  $30. 
June  28,  1914,  $275. 
How  much  was  due  Jan.  16,  1915? 
Ans.,  $620.64. 

3.  Mr.    Rawlins  gave  a  note  for 


$1400  Oct.  12,  1912,  with  interest 
at  6  per  cent  payable  semi-annually. 
He  paid  the  interest  and  $250  on 
the  principal  at  each  interest  pay- 
ment. How  much  was  each  pay- 
ment and  how  much  remained  due 
Oct.  12,  1914?  Ans.,  $271,  $267.25, 
$263.50,  and  $659.75. 

Merchant's  Rule.  Find  the  amount 
of  the  principal  from  its  date  to  the 
time  of  settlement. 

Find  the  interest  on  each  payment 
from  the  time  it  was  made  till  the 
time  of  settlement. 

From  the  amount  of  the  principal 
subtract  the  amount  of  the  payments 
and  their  interest. 


Model. 
Each  date   is   subtracted   from   the 
last. 

A  note  for  $500  dated  May  15, 
1907,  has  the  following  endorse- 
ments: July  10,  1907,  $145;  Oct.  16, 
1907,  $175.  How  much  was  due 
Jan.  1,  1908,  interest  at  6  per  cent? 


Yr. 

Mo. 

Da. 

1908  — 

1  — 

1  .... 

1907  — 

10  — 

16  $175 

1907  — 

7  — 

10  $145 

1907  — 

5  — 

15  .... 

.037%- 

7  — 

16  ....V 

.0285  — 

5  — 

21  $145 V 

.0125  — 

2  — 

15  $175 V 

$500  X 

.0372/3= 

=  $18,833 

$145  X 

.0285  = 

=   4.132 

$175  X 

.0125  = 

=   2.187 

Rate  6% 


Prin. 
Int. 

1st  Pay. 
Int. 

2nd  Pay. 
Int. 

Bal. 

500 

18 

83 

$518 

326 

|$192 

83 

145 
4 

175 
2 

i3 

19 

32 
51 

In  connection  with  this  subject 
discuss  notes,  indorsements,  life  of 
note,  etc. 

1.  A  note  for  $975,  dated  April 
16,  1913,  and  bearing  interest  at 
6%  per  cent  was  indorsed  as  fol- 
lows: 

June  20,  1913,  $375. 

Sept.  16,  1913,  $450. 


How  much  was  due  Mar.  16, 
1914?     Ans.,   $175.46. 

2.  On  a  note  for  $1200,  dated 
July  1,  ]913,  bearing  interest  at  7 
per  cent  payments  of  $300  weit 
made    at    the    end    of    each    quarter. 

How   much   was   due   July   1,   1914? 
Ans.,  $352.50. 


BANK  DISCOUNT 


91 


Bank  Discount. 

Commercial  banks  make  loans  ior 
not  to  exceed  six  months,  as  a  rule, 
with  interest  payable  quarterly  or 
semi-annually.  The  borrower  re- 
ceives the  face  of  the  note  and  pays 
the  face  and  accrued  interest.  Some 
banks  collect  the  interest  in  advance, 
but  this  is  not  customary  in  Califor- 
nia. 

Commercial  paper  issued  by  large 
manufacturing  firms  or  packing  com- 
panies and  county  or  city  warrants 
are  discounted  by  deducting  the  in- 
terest on  the  face  from  the  time  the 
paper  is  purchased  by  the  bank  to 
the  time  it  is  due.  Notes  are  sel- 
dom bought  in  this  way. 

If  an  interest-bearing  note  is  dis- 
counted the  discount  is  reckoned  on 
the  amount  that  will  be  due  on  the 
note  at  the  time  it  falls  due. 

Find  the  discount  by  counting  the 
actual  number  of  days  and  using 
360  days  for  a  year.  Days  of  grace 
are   not  allowed   in   California. 

This  interest  on  notes  or  other  pa- 
per discounted  is  called  bank  dis- 
count, and  the  amount  paid  the  hold- 
er is  the  proceeds. 

Find  the  bank  discount  and  pro- 
ceeds of  a  warrant  for  $220  due  in 
75  days,  and  discounted  at  one  per 
cent  a  month. 

Solution. 
Face  $220 

Discount  5 .  50 


Proceeds  $214.50 

Side  work. 
75  da.  equal  %  mo. 
$2.20xy2    equal   $5.50 

1.  A  time  draft  for  $167.50  due 
in  4  months  is  discounted  at  7  per 
cent.  Find  the  discount  and  pro- 
ceeds . 

2.  Mr.  Anderson  holds  a  90  da. 
time  draft  for  $1062.80,  dated  Aug. 
15.  He  sells  it  Oct.  5  to  the  First 
Nations  Bank  which  discounts  it  at 
6   per   cent.    Find   the   proceeds. 

3.  Mr.  Ryan  holds  a  note  for 
$975.60  due  in  5  months  and  bear- 


ing 5  per  cent  interest.  He  sells  it 
to  the  Commercial  Bank  at  8  per 
cent  bank  discount.  Find  the  pro- 
ceeds . 

4.  A  note  for  $4760  dated  July 
1st,  1914,  and  bearing  interest  at  6 
per  cent  and  due  6  months  after 
date  was  discounted  Sept.  10  at  7 
per   cent.    Find  the   proceeds. 

It  is  sometimes  required  to  find 
the  face  of  a  note  which  will  yield  a 
given  amount  when  discounted  at  a 
bank,  tho  such  a  problem  would 
seldom  if  ever  arise  in  practise.  Ta 
solve  such  a  problem  find  the  pro- 
ceeds of  one  dollar  and  divide  the 
given   proceeds   by   it. 

Find  the  face  of  a  90  day  note 
which  will  give  proceeds  of  $1200 
when  discounted  at  a  bank  at  8  per 
cent. 

Solution. 

Int.  on  $1  for  90  da.  at  8  per 
cent  equals    .02. 

$1.00— $.02   equal   $.98. 
12  24.89 


,98)$1200.00'00 
98 

220 
196 

240 
196 


440 
392 


880 
784 

96 
Ans.    $1224.49. 

1.  Mr.  Watson  has  purchased  an 
automobile  for  $1360.  If  the  bank 
charges  7  per  cent  discount  for  how 
must  a  60  day  note  be  drawn  in  or- 
der to  secure  the  purchase  price? 

2.  Mr.  Peart  has  bargained  for 
an  orchard  for  $22500.  He  has 
$18000  cash  and  is  to  pay  the  bal- 
ance in  four  months.  For  how  much 
must    his    note    be    drawn    that    the 


92 


ARITHMETIC 


balance  may  be  realized  if  the  note 
is  discounted  at  a  bank  at  6^4  per 
cent? 

Present  Worth. 

If  a  sum  of  money  is  to  be  paid  at 
a  future  date  its  present  value  would 
be  the  principal  which  would  amount 
to  that  sum  at  the  given  time. 

1.  What  is  the  present  value  of 
$800  due  in  9  months  without  in- 
terest, money  being  worth  6  per 
cent? 

One  dollar  amounts  to  $1,045  in 
9  months.  $800  divided  by  $1,045 
will  give  the  desired  principal  which 
is  $765.55. 

2.  A  bond  of  $1000  due  in  three 
years  bears  6  per  cent  interest  pay- 
able annually.  If  money  is  worth  5 
per  cent  what  is  the  present  value 
of  the  bond? 

There  will  be  three  payments,  one 
of  $60  in  1  year,  another  of  $60  in 
two  years,  and  a  third  of  $1060  in 
three  years.  The  present  value  of 
the  bond  is  the  sum  of  the  present 
values  of  these  three  payments.  Us- 
ing compound  interest  which  is 
proper  when  payments  are  made  at 
stated  intervals  the  answer  is 
$1027.23. 

3.  Mr.  Crittenden  has  his  farm 
leased  for  five  years  at  a  cash  ren- 
tal of  $5000  a  year  payable  at  the 
end  of  the  year.  What  is  the  pres- 
ent worth  of  the  lease  reckoning 
money  worth  6  per  cent  simple  in- 
terest?    Ans.,  $2196.97. 

In  the  following  problems  use 
compound  interest  making  use  of  a 
table. 

4.  If  money  is  worth  4%,  what 
should  be  paid  for  a  bond  for  $1,000 
payable  in  5  years,  and  bearing  5% 
interest  payable  annually?  Ans., 
$1,044.51. 

5.  Mr.  Brown  bought  a  house 
for  $4,000.  He  is  to  pay  for  it  in 
five  equal  annual  payments,  each 
payment  to  be  made  at  the  end  of 
the  year.    Interest  at  6%  is  allowed 


on  unpaid  balances.   What  is  the  an- 
nual payment?    Ans.,  $949.58. 

6.  Mr.  Clements  proposes  to 
place  $100  in  the  bank  to  the  credit 
of  his  son  each  birthday  from  the 
16th  to  the  21st,  inclusive.  The  bank 
allows  4%  interest  compounded  semi- 
annually. How  much  will  the  son 
have  to  his  credit  when  he  is  twen- 
ty-one?    Ans.,  $663.96. 

7.  Mr.  Moore,  who  is  60  years 
old,  wishes  to  deposit  sufficient 
money  in  a  savings  bank  to  meet 
the  annual  payment  of  $60  on  his 
insurance  policy  for  ten  years,  the 
payments  to  be  made  at  the  end  of 
each  year.  The  bank  allows  4% 
interest  compounded  semi-annually. 
How  much  must  Mr.  Moore  de- 
posit?    Ans.,  $485.68. 

8.  Mr.  Simpson  offers  his  farm 
for  $5,000  cash,  or  $6,060  payable  in 
three  equal  annual  payments  with- 
out interest,  the  payments  to  be 
made  at  the  end  of  the  year.  It 
money  is  worth  6%,  which  proposi- 
tion is  the  best,  and  how  much? 
Ans.,  $5,000  is  $346.02  better  for  the 
purchaser, 

MENSURATION. 

Areas  and  volumes  of  rectangular 
figures  offer  no  serious  difficulty 
and  may  be  presented  in  an  ele- 
mentary way  as  early  as  the  third 
year,  thus  furnishing  excellent  con- 
crete material  for  the  application  of 
multiplication  and  division.  Care 
should  be  taken  that  the  pupil  does 
not  form  the  habit  of  saying  and 
thinking  that  feet  multiplied  by  feet 
give  square  feet,  inches  multiplied  by 
inches  give  square  inches,  square 
feet  multiplied  by  feet  give  cubic 
feet,  square  inches  multiplied  by 
inches  give  cubic  inches,  and  so  on. 
These  notions  arise  from  the  fact 
that  a  rectangle  4  inches  by  5  inches 
for  example  contains  4  times  5  or 
20  square  inches  and  a  rectangular 
solid  3  inches  by  4  inches  by  5 
inches  for  example  contains  3  times 
4  times  5  cubic  inches. 


MENSURATION 


93 


It  is  best  at  first  to  associate  men- 
suration work  with  the  setting  out 
of  trees  or  vines  in  rows  to  form  an 
orchard  or  vineyard,  and  the  pack- 
ing of  eggs  and  fruit  in  boxes,  of 
canned  goods  and  other  packages  in 
cases,  and  similar  work.  A  set  of 
pasteboard  forms  used  for  keeping 
the  eggs  in  a  box  separate  will  be 
very  suggestive.  It  will  be  seen  that 
each  row  will  hold  6  eggs  and  that 
there  are  6  rows,  hence  one  form 
will  hold  6  times  6  eggs,  or  36  eggs. 
Each  of  these  forms  makes  a  layer 
in  one  end  of  an  egg  box  and  there 
are  five  layers  in  the  box  placed  one 
above  another.  An  egg  box  there- 
fore contains  2  times  5  times  36  or 
360  eggs.  How  many  dozens?  In 
like  manner  a  case  of  canned  corn 
contains  2  layers  of  cans  placed  in 
rows  of  3  by  4. 

Have  pupils  cut  two  differnt  col- 
ors of  card  board  into  square  inches. 
These  may  be  built  into  rectangles 
of  different  shape.  The  pupils  will 
soon  be  able  to  determine  that  a  rec- 
tangle 5  inches  by  6  inches  for  ex- 
ample will  contain  5  times  6  or  30 
square  inches.  They  can  also  de- 
termine beforehand  that  it  requires 
21  square  inches  to  make  a  rect- 
angle 3  inches  by  7  inches.  How 
many  square  inches  are  required  to 
make  a  square  foot?  How  many 
square  feet  to  make  a  square  yard? 
Why?  If  square  inches  of  card- 
board are  cut  diagonally  into  tri- 
angles, many  pleasing  patterns  may 
be  made.  Try  such  exercises  with 
children  of  the  third  and  fourth 
years  and  note  the  results. 

The  school  should  be  supplied  with 
cubic  inch  blocks  of  wood.  These 
may  be  used  in  building  up  rect- 
angular solids,  and  thru  this  work 
the  pupil  will  learn  how  to  determine 
the  number  of  cubic  inches  in  a  solid 
of  given  dimensions.  Do  not  teach 
rules  for  mensuration  before  the 
work  is  regularly  taken  up  in  the 
eighth  year. 

1.  Draw  a  rectangle  whose  sides 
are  respectively  4  in.  and  5  in.  Draw 
lines  dividing  it  as  shown  in  Fig.  39. 


■*■ 


CHART    1. 

Note  that  the  figure  is  divided 
into  squares  one  inch  on  a  side,  i.  e., 
square  inches,  that  there  are  five 
squares  in  each  horizontal  row;  that 
there  are  four  such  rows ;  that  there- 
fore the  area  is  4  X  5  sq.  in.,  or  20 
square   inches. 

In  like  manner  show  by  diagram 
that 

144  sq.  in.  =  1  sq.  ft. 
9  sq.  ft.  =  1  sq.  yd. 
30^4  sq.  yd.  =  1  sq.  rd. 

To  find  the  area  of  a  given  sui- 
face  is  to  find  the  number  of  squares 
of  a  given  kind  that  it  contains. 

From  the  problems  given  above, 
it  will  be  seen  that  to  find  the  area 
of  a  rectangle — Multiply  the  number 
of  units  in  the  length  by  the  num- 
ber of  units  in  the  width. 

9.  What  is  the  area  in  sq.  m.  of 
a  square  whose  side  is  27  m. 

10.  Find  the  area  in  sq.  cm.  of 
the    surface    represented    in    fig.    40. 


Seem 


11.     Find   the  area   in   sq.   cm.   of 
the  surface  represented  in  fig.  41, 


c 

e  cm 

3B   crn 


94 


ARITHMETIC 


CARPETING. 

Carpet  is   usually  3   ft.,  or  27   in. 

wide,    and    a   yard   of   carpet  means 

one   yard   in    length   without  regard 
to  the  width. 

Strips. — In  finding  how  many 
yards  of  carpet  are  required  it  is 
necessary  to  decide  which  way  the 
strips  are  to  be  laid  and  then  to 
determine  how  many  strips  are  re- 
quired. A  fractional  part  of  a  strip 
must  be  reckoned  as  a  whole  strip. 

Matching. — Most  carpet  has  a  well 
defined  pattern  which  should  be 
matched  along  the  edges  of  the 
strips.  To  do  this  it  is  necessary 
that  each  strip  shall  begin  at  the 
same  point  of  the  pattern.  Hence, 
in  determining  the  length  of  a  strip 
full  pattern  lengths  must  be  reck- 
oned  on  all  strips   after  the   first. 

Border. — Sometimes  a  carpet  is 
surrounded  with  a  border.  This  bor- 
der   is    matched    at    the    corners    by 


beveling  and   its   length   must    equal 
the  distance  around  the  room. 

1.  A  room  13'  x  16'  is  to  be  cov- 
ered with  carpet  27'"  wide,  the  strips 
to  run  lengthwise.  How  many 
yards  are  required  if  there  is  no 
waste  in  matching? 

2.  How  many  yards  of  carpet 
one  yard  wide  are  required  to  cover 
a  room  20'  x  22',  the  strips  running 
lengthwise,  if  8"  arc  lo^t  in  match- 
ing all  strips  except  the  first? 

3.  A  room  14'8"  x  16'G"  is  to 
be  covered  with  carpet  27"  wide  at 
$1.40  a  yard.  The  strips  are  to  run 
lengthwise  and  there  is  a  12  inch 
pattern.  How  much  will  the  car- 
pet cost? 

4.  A  room  16'  x  18'  is  to  be  car- 
peted with  carpet  27"  wide  at  $1.50 
a  yard  surrounded  with  a  border  16" 
wide  at  $1.35  a  yard.  The  carpet 
has  a  15  inch  pattern  and  the  strips 
are  to  run  lengthwise.  Find  the 
cost  of  the  carpet  and  border. 


Papering  and  Plastering. 

b 

5  cm 

e  cm 

scm                            e  cm 

CHART    II. 


Take  a  box  2>^  in.  long,  2  in. 
wide  and  V/i  in.  deep  without  top. 
Cut  along  the  edges  and  spread  out 
as  shown  in  figure  42.  This  will 
represent  the  walls  and  ceiling  of  a 
room. 

Paper  is  sold  by  the  double  roll 
sixteen  j-^ards  long,  or  the  single 
roll  of  eight  yards,  the  width  being 
18  inches  as  a  rule.  Border  is  sold 
by  the  running  yard. 

There  is  no  fixed  rule  for  allow- 
ance for  opening.  Some  contract- 
ors   deduct    one    half    the    openings. 


others  allow  2  square  yards  for  each 
single  opening.  In  practise  paper- 
hangers  take  sufficient  paper  from 
the  stores  with  the  privilege  of  re- 
turning unused   rolls. 

Plastering  is  done  by  the  square 
yard  and  the  allowance  for  openings 
is  the  same  as  for  papering. 

1,     Find  the  cost  at  $.25  a  double 

roll    for   paper   and   border  at   3c   a 

yard      for     a      room      18'  x     22', 

9'6"  high,  there  being  2  windows 
and  2  doors. 


LUMBER  MEASURE 


95 


2.  A  house  24'  x  30',  witxh  walls 
9'  is  divided  into  four  rooms.  There 
are  ten  windows,  two  outside,  and 
five  inside  doors.  Find  the  cost  of 
plastering  at  27  cents  a  square  yard. 


3.S  m. 


o.  Figure  43  represents  the  floor 
plan  of  a  room  20'  x  26',  the  L 
ends  each  being  12',  height  of  ceil- 
ing 9'.  There  are  six  windows  3'6" 
X  5'2",  and  two  doors  3'  x 
6'8".  Find  the  cost  of  plastering 
the  room  at  30  cents  a  square  yard 
allowing    for   half    the    openings. 

4.  Find  the  cost  at  $1.25  a  yard 
of  carpeting  the  room  described  in 
problem  3  with  carpet  a  yard  wide 
containing  a  16  inch  pattern,  the 
strips  running  lengthwise. 

Lumber    Measure. 

Lumber  is  sold  by  the  thousand 
board  feet  except  that  moldings  and 
the  like  are  sold  by  the  running  foot. 

A  board  foot  is  a  square  foot  of 
lumber  one  inch  thick.  A  thickness 
less  than  an, inch  is  counted  as  an 
inch. 

Thicknesses  of  more  than  an  inch 
are  reckoned  in  inches  and  quarters 
of  an  inch.  When  lumber  is  sur- 
faced on  one  side,  or  sized,  its  thick- 
ness is  reduced  one  eighth  of  an 
inch,  but  the  full  thickness  is  used 
in  calculating  the  amount  of  lumber. 

In  tongue  and  grooved  lumber, 
such  as  flooring,  a  3  inch  board  cov- 
ers 2H  inches  of  floor  space,  a  4 
inch  board  covers  3^  inches.  In 
calculating  the  amount  of  3  inch 
flooring  required  for  a  building,  find 
the  amount  of  floor  space  and  add 
y^  of  it.     For  a  4  inch  flooring  add 

Rustic  and  other  siding  overlaps 
about   an    inch.     Hence,    for   8   inch 


rustic  find  the  number  of  square  feet 
in  the  walls  to  be  covered  and  add 

In  calculating  the  amount  of  lum- 
ber change  the  length  and  width  to 
feet  and  the  thickness  to  inches. 

Find  the  cost  of  36  pieces  of  red- 
wood 2"  X  10"  by  32'  long  @  $22 
M. 

Solution. 
36X2X8_=192  board  feet  = 
3 
.192  xM. 

,192 
22 


384 
384 


4,224 

Ans,  $4.22. 

1.  Find  the  cost  of  the  following 
bill  of  lumber  at  $21  M.  Make  out 
bill. 

28  pieces  2"    x  12",  34'  long. 

48  pieces  1"     x     8",  16'  long. 

20  pieces  13^  "x  10",  14'  long. 

16  pieces  l^"x     6",  18'  long. 

2.  Find  the  cost  at  $32  M.  of  8 
inch  rustic  to  cover  a  building 
36'  X  54'  outside  measurement,  the 
walls  being  14'  high. 

3.  Find  the  cost  at  $38  M.  of  3 
inch  flooring  for  a  one  story  L 
shaped  building  30'  x  36'  with  L 
ends  each  18'  wide. 

4.  Find  the  cost  at  lie  a  foot  for 
chalk  molding  and  6c  a  foot  for  up- 
per molding  for  a  room  33'  x  35' 
with  two  doors  each  4'  wide  and  six 
windows  each  3'6"  wide. 

Shingling. 

Shingles  are  put  up  in  bundles  of 
250  shingles  each  and  sold  by  the 
thousand,  only  whole  bunches  being 
sold. 

While  practically  shingles  are  of 
varying  widths  a  shingle,  commer- 
cially considered,  is  four  inches  wide. 
Shingles  are  usualy  laid  with  4  inch- 
es   or    4^^    inches    exposed.      When 


96 


ARITHMETIC 


laid  4  inches  to  the  weather  one 
shingle  covers  16  square  inches,  or 
9  shingles  cover  a  square  foot.  When 
4^  inches  are  exposed  8  shingles 
cover  one  square  foot. 

Find  the  cost  at  $2.40  M.  of  shin- 
gles laid  4^  in.  to  the  weather  to 
cover  a  roof  each  side  of  which  is 
24'  X  60'. 

Solution. 

2X24X60X8  equal  23040. 

23040  shingles  equal  23.04  M. 

23.25  M.  required. 

23.25  X  $2.40  equal  $55.80    Ans. 

1.  Find  the  cost  at  $2.50  M.  of 
shingles  for  a  roof  each  side  of 
which  is  23'  x  41'  the  shingles  being 
laid  4"  to  the  weather. 

Land  Measure. 

For  a  full  description  of  land 
measure  in  the  United  States  see 
the  California  Advanced  Arithmetic, 
pages   297-303. 

A  township  is  six  miles  square 
and  contains  36  sections  of  640  acres 
each  if  full  size. 

The  sections  of  a  township  are 
numbered  as  shown  in  the  diagram. 


6 

5 

4 

3 

2 

1 

7 

8 

9 

iO 

II 

12 

18 

17 

16 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

1 

32 

33 

34 

35 

36 

When  land  is  surveyed  by  the 
government,  posts  or  stones  are 
placed  along  section  and  township 
lines  every  half  mile,  thus  establish- 
ing section  and  quarter  section   cor- 


ners. An  interior  quarter  section 
comer  must  be  located  at  the  inter- 
section of  lines  joining  opposite 
quarter  section  corners.  A  section 
may  be  further  subdivided  and  the 
portions  described  as  shown  below. 


A 

B 

F 
E 

D 

C 

A. 
B. 
C. 
D. 
E. 


W.  3^  of  sec. 
N.E.   J4  of  sec. 
E.  Yz  of  S.E.  34  of  sec. 
S.W.   M  of  S.E.   Ya  of  sec. 
S.  ^  of  N.W.    Ya  of  S.E. 
Ya  of  sec. 

In  determining  the  amount  of  land 
in  a  portion  of  a  section  deal  with 
the  squares  as  units  bearing  in  mind 
that  a  section  has  640  acres,  a  quar- 
ter section  160  acres  and  a  quarter 
of  a  quarter  40  acres,  etc.  F  in  the 
diagram,  for  example,  has  Y^  of  40 
A.  or  20  A. 

In  following  examples  locate  the 
land  in  the  section  and  the  sections 
in  the  township. 

1.  Find  the  value  of  W.)/^  of 
N.E.  Ya  S.W.M  of  sec.  26  at  $65 
an  acre. 

2.  Find  at  $2.25  a  rod  the  cost  of 
fencing  N.E.  Ya  of  N.W.  Ya  of 
S.W.    Ya  of  sec.  17. 

3.  Mr.  Wallace  purchased  the 
S.E.  Ya  of  sec.  21,  the  W.  Y^  ot 
S.W.  Ya  of  sec.  22,  and  the  N.W. 
Ya  of  N.W.  Ya  of  sec.  27,  all  in  T. 
4  S.,  R.  2  W.  at  $85  an  acre.  Find 
the  cost  of  the  land  and  the  number 
of  rods  of  fence  required  to  enclose 
it 


SQUARE   ROOT 


97 


Rectangles,  Squares. 


In  the  figures  in  tliis  chapter,  use 
the  same  lengths  for  a,  b,  and  c  res- 
pectively thruout. 

1.  By  the  square  on  a  line  is 
meant  the  square  of  which  that  line 
is  one  side. 

The  square  on  the  line,  a,  is  the 
square  which  has  a  for  one  side, 
(Fig.  70).  It  is  written  a^,  and  is 
read,  The  square  on  the  line  a. 

2.  The  rectangle  of  two  lines 
means  the  rectangle  whose  length  is 
one  of  the  lines  and  whose  width 
is  the  other. 


The  rectangle  of  the  lines,  a  and 
b,  is  the  rectangle  whose  length  is 
a  and  whose  width  is  b,  (Fig.  71). 
It   is   written   rect.   ab.   and   is   read 


a 

Vi 

^ 

c* 

i 

^ 

1%. 


Fi 


13 


The  rectangle  of  the  lines  a  and  b, 
or   Rectangle   ab. 

3.  Find  the  sum  of  the  lines,  a 
and  b,  and  construct  a  square  on  this 
sum,  (Fig.  72).  Divide  it  by  draw- 
ing lines  as  shown  in  the  figure.  Cut 
this  out  and  cut  out  another  of  ex- 
actly the  same  size  and  shape.     Cut 


(a+b)2=a2+b2+2  rect.  ab 

the  second  figure  along  the  inside 
lines.  Arrange  the  figures  as  shown 
in  Figure  73. 

The  square  on  the  sum  of  a  and  b 
equals  the  square  on  a,  plus  the  square 
on  b,  plus  twice  the  rectangle  of  a 
and  b. 


Building  a  Square. 


1.     Find  the  square  of  84. 

84  =80-f  4 
802=6400 
2X80X4  =  640 
42=    16 

842=7056 


60 

■4 

SO 

60 

§ 

4 

J^'i^S/. 


98 


ARITHMETIC 


2.     Find  the  square  of  763. 
763=^=700+60+3 

7002=490000 

2X700X60=  84000 

602=  3600 


7602=577600 

2X760X3  =    4560 

32=  9 


7632=582169 


70c 


700 


7eo 


eo 


Note  that  if  a  number  has  one 
cipher  on  the  right  hand  its  square 
will  have  two,  if  it  has  two,  its 
square  will  have  four.  In  general 
the  square  of  a  number  will  have 
twice  as  many  ciphers  on  its  right 
as  the  number  itself  has. 

Square  Root. 

The  building  up  of  squares  sug- 
gests a  method  for  extracting  the 
square  root  of  numbers. 


TOO 

The  two  right  hand  digits  of  a 
whole  number  are  not  used  in  find- 
ing the  tens  digit  of  its  square  root, 
four  are  not  used  in  finding  the 
hundreds,  etc. 

If  a  number  is  marked  off  into 
periods  of  two  figures  each  begin- 
ning at  the  decimal  point  the  figure 
to  be  used  in  finding  the  successive 
figures  of  its  square  root  beginning 
at  the  left  will  be  determined. 


Extract  the  square  root  of  9216. 
9  6 


6 

do 

90 

92'16 

90-= 

=  81  00 

=  180 
6 

11  16 

186 

11  16 

ri^-  8^' 


Explanation. 

The  two  right  hand  figures  are 
cut  off  because  the  square  of  tens 
has  two  ciphers  to  the  right.  The 
square  of  the  tens  then  in  this  case 
must  not  exceed  92,  The  tens  fig- 
ure is  9,  for  9  squared  is  81  the 
largest   square  not  exceeding  92. 

Subtracting  the  square  of  9  tens, 
which  is  8100,  from  9216  leaves 
1116. 


SQUARE   ROOT 


99 


The  90  squared  is  represented  by 
the  upper  diagram,  and  there  is  a 
remainder  of  1116  square  units  to 
be  added  to  it.  In  order  to  increase 
this  square  and  still  have  a  square, 
two  rectangles  each  90  long  and  a 
square  must  be  added. 

The  two  rectangles  have  a  com- 
bined length  of  180,  and,  since  these 
comprise  the  greater  part  of  what 
must  be  added,  180  may  be  used  as 
a  trial  divisor.  1116  divided  by  180 
gives  6  as  the  trial  figure  and  prob- 
able length  of  one  side  of  the  small 
square. 

Placing  the  square  along  with  the 
rectangles  gives  a  total  length  of 
186.  Multiplying  this  length  by  6 
gives  1116,  which  is  the  number  of 
square  units  that  were  remaining 
to  be  added.  Hence  the  size  of  the 
completed  square  is  90  plus  6,  or 
96. 

1.  Find  the  square  root  of  (1) 
7225;  (2)  5329;  (3)  4489;  (4) 
1444;  (5)  7396. 

The  work  may  be  shortened  by 
omitting  the  ciphers  as  shown  be- 
low. 

Find  the  square  root  of  8836. 
9    4 


Extract       the      square 
762129. 

8    7    3 


root      of 


88'36 
81 

184  7  36 
7  36 

2. 
6084; 

Find  the  square 
(2)  5625;  (3) 

root  of  (1) 
4761 ;  (4) 
6561;  (5)  729,  using  the  short  meth- 
od. 

The  same  method  is  used  when 
the  answer  contains  more  than  two 
figures. 


76'21'29' 
64 

167  1  12  21 
1  1169 

1743 

52  29 
52  29 

3.  Find  the  square  root  of  (1) 
245025;  (2)  375769;  (3)  249001; 
(4)   529984. 

4.  Square   •% ;    ^fis ;     %5;    % ; 

5.  Find  the  square  root  of  % ; 
«yi2i;  121/196;  ^%84;  2%; 

421/25;  2%9;262%,. 

6.  Square  .3;  .7;  .02;  .67; 
.009;  .0468. 

Compare  the  number  of  decimal 
places  in  the  number  with  the  num- 
ber of  decimal  places  in  its  square. 

7.  Find  the  square  root  of  (1) 
.8649;  (2)  51.84;  (3)  .0025;  (4) 
94.2841;  (5)  .000529;  (6)  .00000- 
23409. 

Sometimes  the  square  root  of  a 
number  can  be  found  only  approxi- 
mately. 

Find  the  square  root  of  38.7. 
6.     2    2 


38. 
36 

'70'00 

J2i2 

|2 

70 
44 

1242  1  26  00 

1  24  84 

In  pointing  off  decimals  begin  at 
the  decimal  point. 

8.  Find  to  two  decimal  places 
the  square  root  of  (1)  24;  (2)  6.4: 
(3)    .76;   (4)    .144:   (5)   9.04. 

9.  Find    in    rods    one    side    of    a 


100 


ARITHMETIC 


square  field  which  contains  (1)  10 
A.;  (2)  22>^  A.;  (3)  62>4  A. 
For  additional  examples  in  square 
root  see  the  Grammar  School  Arith- 
metic, California  State  Series,  pages 
327-329. 


i L t 

I                       I  > 
I                       < 

I  > 

> 

I                      >  I 

> 

I                      '  I 

I                       '  ■ 

I                       I  I 
*                       I 

I  I 

I                       I  I 

I 1 _.   '      _ 1 


Draw  a  rectangle  whose  width  is 
to  its  length  as  3  is  to  4.  Divide  it 
as  shown  in  fig.  85.  Into  how  many 
parts  is  the  figure  divided?  What 
is  the  shape  of  each  part?  If  the  en- 
tire figure  contains  588  sq.  mm.  what 


is  the  length  of  one  side  of  one  of 
the  small  divisions?  Find  the  width 
and  length  of  the  figure. 

10.  Find  the  dimensions  in  rods 
of  a  field  containing  4^  A.,  whose 
width  is  five-ninths  of  its  length. 

11.  Find  the  perimeter  in  rods  of 
a  field  containing  36  A.,  whose  width 
is  .9  of  its  length. 

12.  The  entire  surface  of  a  cub- 
ical block  is  2646  sq.  cm.  What  is 
the  length  of  one  side? 

13.  The  length,  breadth,  anl 
thickness  of  a  block  are  in  the  ratio 
of  4,  3,  and  2,  and  its  entire  sur- 
face is  1300  sq.  cm.  Find  its  di- 
mensions. 

14.  The  length,  breadth,  and 
thickness  of  a  solid  are  in  the  ratio 
of  5,  3,  and  2,  and  its  entire  surface 
is  7502  sq.  in.  Find  the  sum  of  all 
its  edges. 


RIGHT  TRIANGLES. 


1.  Take  the  rectangle  ab,  and 
draw  the  diagonal  c.  This  will  di- 
vide the  rectangle  into  two  right 
triangles  with  base,  a;  perpendicu- 
lar, b;  and  hypotenuse,  c.  Draw 
the  square  on  c.  Cut  out  and  paste 
as  shown  in  figures  76  and  77. 

2.  Cut  out  a  square  on  a-|-b,   a 


Fiq  ^ 


<^  11 

square  on  c,  and  four  right  tri- 
angles with  sides,  a,  b,  and  c.  Ar- 
range and  paste  as  shown  in  figure 
78. 

Are  the  outside  lines  of  the  right 
hand  figure  straight?  How  long 
are  they?  Compare  the  triangles 
with  the  rectangles. 


RIGHT  TRIANGLES 


101 


a 

b 

a 

b 

a24-b2+2  rect.  ab  =  c2+2  rect.  ab. 


r— —  - 

h 
b 

a 

1 
1 

1 

1 
1 
I 
1 

1 
1 
1 
1 

! 

1 

a^  -{-  b^  =  c^ 


3.  Remove  the  rectangles  and  tri- 
angles as  shown  in  Fig.  79.  Com- 
pare the  remainders. 

Arrange  the  figures  as  shown  in 
fig.  80,  and  paste  them  in  your  book. 
The    dotted    lines    show    where    the 


pieces  were  removed. 

This  proves  that — The  square  on 
the  hypotenuse  of  a  right  triangle 
equals  the  sum  of  the  squares  on  the 
other  two  sides. 


4.  Find    the    distance    diagonally 
across  a  room  20  by  30  ft. 

5.  Find     the     shortest     distance 


from  a  lower  to  the  opposite  corner 
of  a  room  16  by  20  ft.  by  14  ft. 
high. 


102 


ARITHMETIC 


A  diagram  will  be  very  helpful  in 
the  solution  of  the  problems  which 
follow.  For  problems  like  that  of 
spider  and  fly,  use  a  pasteboard  box. 
Cut  along  the  corners  and  lay  sides 
down  on  a  level  with  the  floor.  The 
solution  will  then  become  evident. 

6.  A  flag-pole  100  feet  high  stood 
16  feet  from  a  square-topped  hen 
house  12  feet  high  and  14  feet  wide. 
The  pole  fell,  struck  the  upper  edge 
of  the  hen  house  and  broke.  The 
lower  part  of  the  remaining  portion 
tilted  up  without  slipping,  and  the 
top  struck  the  ground.  How  far 
from  the  base  did  the  top  of  the 
pole  strike?     Ans.,  94.89  ft. 

7,  A  spider  is  on  the  floor  three 
feet  from  one  wall  of  the  room  and 
six  feet  from  another.  A  fly  is 
caught  in  a  web  in  the  nearest  up- 
per corner,  fourteen  feet  above  the 
floor.  Find  the  shortest  distance 
that  the  spider  must  travel  along 
the  floor  and  wall  to  reach  the  fly. 


Ans.,  18.02  ft. 

8.  A  room  is  20  X  30  feet  and 
15  feet  high.  Find  the  shortest  dis- 
tance along  the  floor  and  walls  from 
a  lower  corner  to  the  opposite  up- 
per corner.    Ans.,  46.09  ft. 

9.  A  log  is  20  feet  long  and  16 
inches  in  diameter.  A  string  is 
wrapped  about  the  log  from  end  to 
end,  each  circuit  being  one  foot  from 
the  others,  the  distance  being  meas- 
ured lengthwise  of  the  log.  How 
long  is  the  string?     Ans.,  86.13  ft. 

10.  Find  the  shortest  distance 
along  the  surface  of  a  block  10X12 
X15  inches  from  one  corner  to  the 
one  directly  opposite.  Ans.,  26.62 
in. 

11.  A  hollow  cylinder  is  6  inches 
in  diameter  and  30  inches  long.  Find 
the  shortest  distance  along  the  sur- 
face from  a  point  on  the  upper  edge 
to  the  opposite  point  on  the  lower 
edge.     Ans.,  31.4  in. 


Farallelogram. 


Chart 

Draw  a  parallelogram  4  in.  wide 
and  5  in.  long  as  shown  in  figure 
44.  Cut  it  out  and  then  using  it  as 
a  pattern  cut  out  another  of  the  same 
size. 

Cut  off  the  corner  of  the  second 
figure  and  place  it  on  the  right  as 
shown  in  Fig.  45. 

From  this  it  is  seen  that — A  par- 
allelogram is  equal  to  a  rectangle  of 


parallel- 
in.    and 


F^i^.  4S. 


III. 

the  same  base  and  height. 

1.  Find  the  area  of  a 
ogram  whose  length  is  12 
width  9  in. 

2.  A  parallelogram  is  8.5  ft.  long 
and  224  in,  wide.  How  many  sq. 
ft.  does  it  contain? 

3.  A  parallelogram  whose  length 
is  24  ft.  has  an  area  of  336  sq.  ft. 
How  wide  is  it? 


Triangle. 


^V^  -=7  0. 


j^i^.^r. 


Chart  IV. 


TRAPEZOID 


103 


Draw  a  triangle  with  base  5  in. 
and  height  4  in.  as  shown  in  Fig. 
46.  Cut  it  out  and  cut  out  two  more 
of  the  same  size.  Place  the  second 
and  third  as  shown  in  Fig.  47. 

Note  that  Fig.  47  is  a  parallel- 
ogram having  the  same  base  and  al- 
titude as  the  triangle  in  Fig.  4G,  but 
twice  the  area. 

From  this  it  is  learned  that — A 
triangle  equals  half  a  parallelogram  of 
the  same  base  and  height. 

Make  a  rule  for  finding  the  area 
of  a  triangle. 

1.  Find  the  area  of  a  triangle 
whose  base  is  52  in.  and  height  28 
in. 

2.  A  triangle  has  a  base  of  8.74 
ft.  and  height  of  1.26  ft.  How  many 
sq.   ft.  in  it? 

3.  Find  the  height  of  a  triangle 
whose  base  is  3.5  yd.  and  area  14 
sq.  yd. 

4.  A  triangle  whose  area  is  275 
sq,  yd.,  is  25  yd.  high.  What  is  its 
base? 

5.  Find  the  number  of  sq.  cm. 
in  the  surface  of  Fig.  48. 

6.  Make  necessary'  measurements 
in  Fig.  49  and  find  the  number  of 
sq.  mm.  it  contains. 


/■•jy.  48- 

In  an  isosceles  or  an  equilateral 
triangle,  the  perpendicular  from  the 
vertex  to  the  base  strikes  the  base 
at  its  midpoint. 

7.  Find  the  altitude  and  area  of 
an  isosceles  whose  equal  sides  are 
each  13  feet  and  base  10  feet.  Ans., 
Area  60  sq.   ft. 

8.  The  perimeter  of  an  isosceles 
triangle  is  100  rd.  and  its  base  is 
40  rd.  Find  the  area.  Ans.,  447.2 
sq.  rd. 

9.  Find  the  altitude  and  area  of 
an  equilateral  triangle  each  of  whose 
sides  is  30  feet.  Ans.,  Area  389.7 
sq.  ft.    • 

10.  The  area  of  an  isosceles  tri- 
angle is  4800  sq.  ft.  and  its  base  is 
120  ft.  Find  the  length  of  one  of 
the  equal  sides.     Ans.,  100  ft. 

If  a  line  is  drawn  from  each  ver- 
tex of  a  regular  hexagon  (six  sid- 
ed figure)  to  the  center,  the  figure 
will  be  divided  into  six  equilater- 
al triangles. 

11.  Find  the  area  of  a  regular 
hexagon  each  of  whose  sides  is  50 
ft.  long.    Ans.,  6495  sq.  ft. 


Trapezoid. 


J^i^.  <TO.  J^l'^.  6'/. 

Chart  V. 

A    trapezoid    has    only   two   of   its      shown   in   Fig.   50 
sides  parallel.  These  are  called  bases. 

Draw  a  trapezoid  whose  bases  are 
5  in.  and  4  in,  and  height  3  in.  as 


Cut  it  out  and 
cut  out  two  more  of  the  same  size. 
Place  the  second  and  third  figures 
as  shown  in  Fig,  51. 


104 


ARITHMETIC 


Note  (1)  that  Fig.  51  is  a  par- 
allelogram, (2)  that  it  has  the  same 
altitude  as  the  trapezoid  in  Fig.  50, 
(3)  that  the  base  of  the  parellogram 
equals  the  sum  of  the  bases  of  the 
trapezoid,  and  (4)  that  the  area  of 
the  parallelogram  equals  twice  the 
area  of  the  trapezoid. 

This  shows  that  —  A  Trapezoid 
equals  half  a  parallelogram  of  the 
same  height  and  having  a  base  equal 
to  the  sum  of  the  base  of  the  trape- 
zoid. 

To  find  the  area  of  a  trapezoid, 
multiply  the  sum  of  its  bases  by  its 
altitude  and  divide  the  product  by 
two. 

1.  Find  the  number  of  sq.  cm.  in 
a  trapezoid  whose  bases  are  12  cm. 
and  10  cm,  and  height  9  cm. 

2.  The  bases  of  a  trapezoid  are 
25  yds.  and  15  yds.  and  the  height 
562  yds.  How  many  sq.  yd.  does 
it   contain? 

3.  The  bases  of  a  trapezoid  are 
15  yd.  and  9  yd.  and  its  area  167 
sq.  yd.     What  is  its  height? 

4.  A  trapezoid  contains  324  sq. 
cm.  Its  height  is  12  cm.  and  upper 
base  24  cm.     Find  the  lower  base. 

5.  Make  the  necessary  measure- 
ments and  find  the  number  of  sq. 
mm.  in  Fig.  52. 


6.  The  roof  of  a  certain  house 
consists  of  two  equal  trapezoids 
whose  lower  and  upper  bases  are  48' 
and  32'  respectively  and  altitudes 
25',  and  two  triangles  whose  bases 
are  each  16'  and  altitudes  25'.  Find 
the  cost  at  $2.75  M.  of  shingles  to 
cover  it.  The  shingles  are  to  be  laid 
4"  to  the  weather. 


7.  The  roof  of  an  L  shaped  house 
consists  of  two  triangles,  two  par- 
allelograms, and  two  trapezoids  each 
with  an  altitude  of  18  feet.  The 
bases  of  the  triangles  are  30',  of  the 
parellograms  14',  and  of  the 
trapezoids  44'  and  14'  respectively. 
How  many  bunches  of  shingles  laid 
^Yz"  to  the  weather  are  required  to 
cover  it? 

Circle. 

Open  your  dividers,  place  one 
point  on  the  paper,  and  revolve  the 
other  about  it  making  a  mark.  The 
figure  formed  is  a  circle,  the  bound- 
ing line  is  the  circumference,  the 
fixed  point  on  which  one  point  of 
the  dividers  was  placed  is  the  cen- 
ter, a  straight  line  drawn  from  the 
center  to  the  circumference  is  a  ra- 
dius, a  line  drawn  thru  the  center 
and  terminated  by  the  circumference 
is  a  diameter. 


The  following  rules  are  proved  in 
geometry. 

Circumference  of  a  circle  equals 
2  X  3.1416  times  the  radius. 

Area  of  a  circle  equals  3.1416 
times  the  square  of  the  radius. 

The  Greek  letter  ir  (pi)  is  used 
in  mathematics  to  stand  for  3.1416 
(3%  nearly).  More  briefly  these 
rules  are  written 

Circum  0=27rr 
Area  O  ==  -ky^ 

1.  Find  the  circumference  of  a 
circle  whose  diameter  is  15  feet. 


VOLUMES 


105 


2.  A  grindstone  is  2  feet  in  diam- 
eter. Find  its  circumference  in  inch- 
es. 

3.  A  tree  is  30  feet  in  circum- 
ference.    P'ind  Its  diameter. 

4.  Find  the  diameter  in  feet  of  a 
circular  race  track  a  mile  in  lengtli. 

5.  Find  the  number  of  square 
inches  of  tin  in  a  6  in.  joint  of  stove- 
pipe 20  inches  long,  making  no  al- 
lowance for  seam. 

6.  Find  the  area  of  a  circular 
grass  plot  16  feet  in  diameter. 

7.  A  horse  is  staked  with  a  rope 
40  feet  long.  Over  how  much 
ground  can  it  graze? 

8.  How  much  lawn  can  be  wat- 
ered with  a  50  foot  hose  attached  to 
a  hydrant,  if  the  water  is  thrown  12 
feet  beyond  the  end  of  the  hose? 

9.  A  galvanized  tank  is  8  feet 
in  diameter  and  7  feet  high.  How 
many  square  feet  of  iron  are  used 
in  making  it? 

10.  Four  equal  circles  10  inches 
in  diameter  are  cut  from  a  piece  of 
pasteboard  20  inches  square.  How 
much  of  the  pasteboard  remains? 
Find  the  area  of  each  piece. 

11.  A  tree  is  three  feet  in  diam- 
eter. Find  the  side  of  a  square  that 
will  have  the  same  area. 

12.  A  horse  is  tied  to  the  corner 
of  a  barn  30  X  40  with  a  rope  70 
feet  long.  Over  how  much  ground 
can  the  horse  graze?  Ans.,  13508.88 
sq.    ft. 

13.  Over  how  much  ground  can 
the  horse  graze  if  tied  to  the  middle 
of  the  longest  side  of  the  same  barn 
with   the   same   rope? 

Ans.,  12252.24  sq.  ft. 

14.  Over  how  much  ground  can 
the  horse  graze  if  tied  to  the  mid- 
dle of  the  shortest  side  of  the  same 
barn  with  the  same  rope?  Ans., 
12802.02  sq.  ft. 

15.  A  rope  is  stretched  between 
two  stakes  25  feet  apart.  A  horse  is 


tied  with  a  rope  30  feet  long  fas- 
tened to  a  ring  which  slips  back  and 
forth  on  the  first  rope.  Over  how 
much  ground  can  the  horse  graze? 
Ans.,  4327.44  sq.   ft. 

16.  A  horse  is  tied  with  a  rope  50 
feet  long  fastened  to  the  top  of  a 
pole  12  feet  high.  Over  hov/  much 
ground  can  the  horse  graze? 

SOLIDS  AND  VOLUMES. 

A  cube  is  a  soild  whose  faces  are 
equal  squares.  (Fig.  53.)  How 
m.any  faces  has  a  cube?  How  many 
edges?  How  many  verticles  (cor- 
ners) ? 


rif  SJ 


If  each  edge  of  a  cube  is  one 
foot,  the  figure  is  a  cubic  foot  (cu. 
ft.).  If  each  edge  is  one  inch,  it  is 
a  cubic  inch   (cu.  in.),  etc. 

1.  Draw  the  surfaces  of  a  cu.  in. 
as  shown  in  Fig.  54.  How  many 
sq.  in.  on  the  surface? 

2.  How  many  sq.  in.  on  the  sur- 
face of  a  cube  whose  edge  is  3  in.? 

3.  If  the  area  of  a  cube  is  150 
sq.  in.,  how  long  is  the  edge? 


y  y  y  .a 


y  77^ 


Z' 

,^ 

y 

y 

y 


2^1  (7-  t5~c57 

4.  Draw  a  cube  whose  edge  is  4 
inches.  Mark  it  as  shown  in  Fig. 
55.  What  is  the  size  of  each  of  the 
small  solids  marked  off?  How  many 
cu.  in.  in  one  row  of  the  top  layer? 


106 


ARITHMETIC 


How  many  rows  in  that  layer?   How  whose  height  is  5  inches.     Paste  the 

many  layers  in  the  whole  cube?   How  sides    and    bottom    and    cut    the    top 

many  cu.  in.  in  the  cube?  off.     How  many  sq.  in.  of  cardboard 

5.  How   many  cu.   in.   in  a  cube  ^^.1  required?      How    many    cu.    in. 
whose  edge  is  6  in.? 

6.  How    many    cu.    in.    in    a    cu. 
ft? 


will  make  one  layer  on  the  bottom? 
How  many  cu.  in.  will  the  prism 
hold? 


PYTiAMW. 


J='Ii/SJyc. 


CrJL/JVUBR. 


7.  How  many  cu.  yd.  of  earth  is 
removed  in  digging  a  cellar  3  yd. 
each  way? 

8.  How  many  cu.  ft.  in  a  cu. 
yd.? 

A  rectangular  solid  is  a  solid  with 
six   rectangular   faces. 

Draw  a  rectangular  solid  whose 
length,  breadth  and  thickness  are  4 
in. ;  3  in.  and  2  in.,  respectively. 
Draw  lines  dividing  it.  Find  the 
sum  of  its  edges.  Draw  its  faces  as 
in  exercise  1.  Find  its  area.  How 
many  cu.  in.  does  the  solid  contain? 

When  we  find  the  number  of 
cubes  that  a  solid  contains  we  say 
we  find  its  volume. 

9.  Find  the  volume  in  cu.  in. 
of  a  solid  4  in.  by  5  in.  by  7  m. 
Find  the  number  of  sq.  in.  in  its 
surface. 

10.  Find  the  volume  in  cu.  ft. 
of  a  solid  25  ft.  by  3  ft.  by  15  yd. 

PRISMS  AND  CYLINDERS. 

A  right  prism  is  a  solid  whose 
bases  are  polygons  and  whose  sides 
are  rectangles. 

Mention  some  objects  that  have 
the  form  of  a  prism.  Construct  of 
cardboard  a  square  prism  whose 
base   is   4   inches   on  each   side,   and 


If  a  rectangle  is  made  to  revolve 
about  one  of  its  sides,  two  of  its 
sides  describe  circles,  and  the  other 
describes  a  curved  surface.  The 
solid  thus   formed  is  a  cylinder. 

Construct  of  cardboard  a  cylinder, 
making  the  base  4  inches  in  diameter 
and  the  height  5  inches,  omit  the 
top. 

To  find  the  volume  of  any  prism 
— Multiply  the  base  by  the  height. 

1.  The  sides  of  a  right  prism  are 
three  feet  by  four  feet,  and  its  height 
is  twelve  feet.  Find  its  volume  and 
its  lateral  area. 

2.  A  hollow  prism  is  three  by 
five  feet  on  the  outside  and  two  by 
three  feet  on  the  inside,  and  it  is 
forty  feet  long.  Find  its  volume 
and  the  area  of  its  outer  and  inner 
lateral  surfaces. 

3.  How  much  sheet  iron  is  used 
in  making  a  six  inch  joint  of  stove 
pipe  thirty  inches  long,  allowing 
four-tenths  of  an  inch  for  the  seam? 

4.  How  many  sq.  cm.  of  tin  are 
required  to  make  a  lard  can  12  cm. 
in  diameter  and  14  cm.  high,  with  a 
lid  whose  rim  slips  down  1  cm.? 
Draw  the  developed  surface.  How 
many  cu.  cm.  of  lard  will  the  can 
hold?  What  will  the  lard  weigh  if 
1  cu.  in.  of  lard  weighs    .84  g.  ? 


FRUSTUMS 


107 


5,     Find  the  weight  of  a  piece  of  Construct    of     cardboard     a     cone 

copper  wire  4  mm.   in  diameter  and  whose  base  is  4  in.  in  diameter,  and 

75  m.   long.      (1   cu.   cm.   of   copper  height  5  in.     Paste  the  sides  of  your 

weighs  8.6  g.).  cone  and  omit  the  bottom. 


COJVE. 

6.  Find  the  weight  of  a  circular 
piece  of  gold  5  cm.  in  diameter  and 
6  mm.  thick.  (1  cu.  cm.  of  gold 
weighs  19.3  g.) 

Pyramids  and  Cones. 

Pyramid — A  solid  whose  base  is  a 
polygon  and  whose  sides  are  tri- 
angles meeting  in  a  common  point  is 
a  pyramid. 

Construct  of  cardboard  a  square 
pyramid  with  the  sides  of  the  base 
each  4  inches  and  perpendicular  its 
height  5  in.  Paste  the  sides  of  the 
pyramid  together  and  cut  off  the 
bottom. 

Fill  the  pyramid  with  sand  and 
then  pour  the  same  into  the  prism 
which  you  made.  Fill  the  pyramid 
again  and  empty  it.  It  will  take 
three  pyramidfuls   to   fill   the   prism. 

To  find  the  volume  of  a  p3rramid — 
Multiply  the  base  by  one-third  of  the 
height. 

7.  How  many  cu.  in.  will  the 
pyramid  which  you  have  made  hold? 

8.  Find  the  volume  of  a  square 
pyramidal  block  of  wood  with  base 
10  in.  square  and  height  12  in. 

Cone — If  a  right  triangle  is  made 
to  revolve  about  one  arm,  the  other 
arm  describes  a  circle,  and  the  hypot- 
nuse  describes  a  curved  surface. 
The  solid  generated  is  a  cone. 


FRUSTUM 

Fill  the  cone  with  sand  and  empty 
into  the  cylinder  of  the  same  dimen- 
sions until  the  cylinder  is  full.  Three 
conefuls  are  required. 

To  find  the  volume  of  a  cone — 
Multiply  the  base  by  the  height  and 
divide  by  three. 

9.  How  many  cu.  in.  in  the  cone 
which  you  have  constructed? 

10.  A  conical  block  of  wood  is 
8  in.  in  diameter  at  the  base  and  16 
in.  long.  How  many  cu.  in.  of  wood 
does   it  contain? 

Fmstnms. 

The  part  of  a  pyramid  or  cone 
contained  between  the  base  and  a 
plane  parallel  to  the  base  is  a  Frus- 
tum. The  mid-base  is  the  base  mid- 
way between  the  upper  and  lower 
bases. 

To  find  the  volume  of  a  frustum — 
Add  together  the  upper  base,  the  low- 
er base,  and  four  times  the  mid- 
base,  multiply  the  sum  by  the  height 
and  divide  the  product  by  six. 

Note — This  rule  is  exact,  and  also 
gives  correctly  the  contents  of  an 
enbankment,  a  ditch,  or  a  frustum  of 
a  wedge. 

To  find  the  lateral  surface  of  a 
frustum — Multiply  the  perimeter  of  the 
mid-base  by  the  slant  height. 

Any  line  of  the  mid-base  is  a  mean 
between  the  corresponding  lines  of 
the  upper  and  lower  bases. 


108 


ARITHMETIC 


Find  the  volume  of  a  frustum 
with  rectangular  bases,  whose  lower 
base  is  18X24  inches,  upper  base 
10  X  16  inches,  and  height  30  inches. 

Solution : 
Length  of  mid-base   (24  in.-|-16  in.) 

^2=20  in. 

Width  of  mid-base    (18  in.+lO  in.) 

-^2=14  in. 

Lower   Base=       24X18=  432 

Upper  Base=       16X10=  160 

4Xmid-base=4X20Xl4=1120 

Sum  1712 

1712X30^6=8560 
Hence  the  volume  is  8560  cu.  in. 

1.  A  dishpan  is  36  cm.  in  diam- 
eter at  the  top,  20  cm.  at  the  bot- 
tom, and  12  cm.  deep.  How  many 
liters  of  water  are  required  to  fill 
it?    Ans.,  7.59  1. 

2.  A  telegraph  pole  is  14  in. 
square  at  the  base,  8  in.  square  at 
the  top  and  48  ft.  long.  How  many 
cubic  feet  does  it  contain?  Ans., 
41.3-f  cu.  ft. 

3.  The  base  of  a  chimney  90  ft. 
high  is  16  ft.  square  on  the  outside 
and  10  ft.  square  on  the  inside.  At 
the  top  the  chimney  is  8  ft.  square 
on  the  outside  and  6  ft.  square  on 
the  inside.  How  many  cubic  feet 
of  masonry  in  the  chimney?  Ans., 
7560  cu.  ft. 

4.  A  round  hollow  iron  pillar, 
18  ft.  long,  is  16  inches  in  diameter 
at  the  base,  and  12  at  the  top.  The 
iron  is  three  inches  thick  at  the  base 
and  two  at  the  top.  How  much  will 
the  pillar  weigh  if  a  cubic  inch  of 
iron  weighs  .26  lb.  Ans.,  5116.5-1- 
Ib. 

A  wedge  may  be  considered  a 
frustum  in  which  the  upper  base  is 
a  line  and  its  area  O.  The  short 
line  of  the  mid-base  is  half  the  cor- 
responding line  of  the  lower  base. 
A  hip  roof  together  with  the  upper 
ceiling  forms  a  kind  of  wedge,  the 
ridge  being  the  upper  base. 

5.  The  base  of  a  wedge  9  in. 
long  is  2  in.  by  3  in.,  the  edge  is  2 
in.  long  and  parallel  to  the  long  side 


of  the  base.     Find  its  volume.     Ans., 
24  cu.  in. 

6.  Find  the  number  of  cubic  feet 
enclosed  by  the  hip  roof  of  a  house 
36  ft.  by  44  ft.,  one-third  pitch. 
Ans.,  6912  cu.  ft. 

One-third  pitch  means  that  the 
heights  of  the  ridgepole  above  the 
upper  wall  plate  is  one-third  of  the 
width  of  the  building.  The  ridge- 
pole of  a  hip  roofed  house  equals 
the  difference  between  the  length 
and  width  of  the  building.  In  the 
building  described  in  the  last  prob- 
lem the  height  of  the  roof  is  12  ft. 
and  the  ridgepole  is  8  feet  long, 

7.  Find  the  volume  of  a  regular 
hexagonal  pyramid  whose  altitude  is 
30  feet  and  each  side  of  whose  base 
is  40   feet.     Ans.,  41568  cu.   ft. 

8.  Each  side  of  the  lower  base 
of  a  frustum  of  a  pyramid  is  8  feet, 
each  side  of  the  upper  base  is  6  ft., 
and  the  altitude  is  12  ft.  Find  the 
volume.     Ans.,   592   cu.    ft. 

9.  A  tank  in  the  form  of  a  frus- 
tum of  a  cone  and  constructed  of 
two-inch  lumber  has  the  following 
outside  measurements :  Height  5  ft. 
8  in.,  diameter  of  base  6  ft.,  diam- 
eter of  top  6  ft.  8  in.  How  many 
gallons  will  it  hold?  Ans.,  1161.5 
gal. 

Levees,  canals,  railway  cuts 
and  embankments  may  be  consid- 
ered as  frustums  lying  on  one  side. 
Their  volumes  are  found  by  calculat- 
ing in  sections  of  50  ft.  or  100  ft. 
in  length.  The  end  sections  of  a 
levee  are  usually  trapezoids. 

10.  Find  the  volume  of  the  fifty 
foot  section  of  a  levee  with  the  di- 
mensions here  given. 

width 


top  I  bottom 


10'  I      18' 
10'  I      22' 
Ans.,  5650  cu.  ft. 


ht. 


Sphere. 

Surface  of  a  sphere  =  4:Trr^ 

Volume  of  a  sphere  =  %7rr3   or   YQird^ 


RATIO 


109 


Note  that  a  sphere  has  just   four 

mes  thf^  arpp  nf  p   rirrle  of  the  same 

radius. 


Note  that  a  sphere  has  just 
times  the  area  of  a  circle  of  the  same 


11.  Find  the  area  of  a  sphere 
whose  diameter  is  1  foot. 

12.  Find  the  radius  of  a  sphere 
whose  area  is  1  sq.  m. 

13.  The  diameter  of  the  earth  is 
7918  miles.  Find  its  circumference 
and  its  area. 

Area,  196961744.4  sq.  mi. 
Circumference,   24875.1888  mi. 

RATIO. 

Ratio  is  the  indicated  quotient 
arising  from  dividing  the  first  of 
two  given  numbers  by  the  second. 
The  ratio  of  6  to  9  is  6-f-9=%,  or 
Yz,  the  ratio  of  12  feet  to  7  feet  is 
12-^7  or  1%.  A  ratio  may  be  in- 
dicated by  the  use  of  a  colon,  as 
5:9,  or  by  a  fraction,  as  %.  In 
either  case  it  is  read  the  ratio  of  5 
to  9.  The  fraction  form  is  prefer- 
able. 

The  first  term  of  a  i^atio  is  called 
the  antecedent  and  the  second  the 
consequent. 

Division,  fraction,  and  ratio  mean 
practically  the  same  thing.  Thus 
ll-f-8,  ii/s,  and  the  ratio  ^%  or  11:8 
are  only  three  ways,  getting  the  same 
results,  lyg.  Note  that  dividend, 
numerator,  and  the  antecedent  are 
corresponding  terms,  as  are  divisor, 
denominator,  and  consequent.  The 
corresponding  values  are  called  quo- 
tient, fraction,  and  ratio  respectively. 

The  following  principles  are  easily 
verified : 

1.  Multiplying  the  dividend  mul- 
tiplies the  quotient  by  the  same  num- 
ber. 

2.  Dividing  the  dividend  divides 
the  quotient  by  the  same  number. 

3.  Multiplying  the  divisor  divides 
the  quotient  by  the  same  number. 

4.  Dividing  the  divisor  multiplies 
the  quotient  by  the  same  number. 

5.  Multiplying    or    dividing  both 


dividend  and  divisor  by  the  same 
number  does  not  change  the  value  of 
the  quotient. 

Restate  these  principles  substitut- 
ing numerator  for  dividend,  denomi- 
nator for  divisor,  and  fraction  for 
quotient. 

Restate     them      substituting  the 

words    antecedent,    consequent,  and 

ratio  for  dividend,  divisor  and  quo- 
tient respectively. 

Note  (1)  that  ratio  can  exist  only 
between  like  quantities,  (2)  that  the 
value  of  a  ratio  is  an  abstract  num- 
ber. 

Feet  and  yards  or  gallons  and 
quarts  can  be  compared  only  by 
changing  to  the  same  denomination. 
Hogs  and  sheep  can  be  compared  by 
weighing  or  valuing.  Fractions  can 
best  be  compared  by  reducing  to  a 
common  denominator  and  comparing 
their  numerators. 

Find  the  ratio  of: 

1.  25  to  45. 

2.  78  to  18. 

3.  4%  to  5^. 

4.  65  lb.  to  39  lb. 

5.  7  ft.  to  28  in. 

6.  19  pt.  to  4  gal. 

7.  2  yd.  to  7  yd.   1  ft. 

8.  4  cords  of  wood  @  $7  to  20 
sheep  @  $3%. 

9.  6748  lb.  prunes  @  l%c  to 
4685  lb.  peaches  @  l^c. 

10.  465  hogs  averaging  225  lb. 
@  7c  lb.  to  160  steers  averaging  525 
lb.  @  lie  lb. 

11.  2%  miles  to  1  mile  167^  rd. 

Proportional  Parts. 

Divide  $17500  between  Charles 
and  Helen  so  that  their  shares  shall 
have  the  ratio  of  3  to  4. 

The  conditions  of  the  problem 
will  be  satisfied  if  the  money  is  di- 
vided into  seven  equal  parts  and 
Charles  is  given  three  and  Helen 
four  parts. 


110 


ARITHMETIC 


Solution. 
Let   3  parts   equal   Charles'   share. 
And  4  parts   equal   Helen's   share. 
Then  7  parts  equal  the  whole  amt. 
7  parts  equal  $17500. 
1  part  equals  $2500. 

3  parts  equal  $7500,  Charles'  share. 

4  parts  equal  $10000,  Helen's  share. 

The  value  of  one  of  the  equal 
parts  may  be  represented  by  some 
letter  as  x. 

The  ages  of  Reuben,  Silas  and 
Joseph  have  the  ratio  3,  4,  and  5, 
and  the  sum  of  their  ages  is  36  years. 
What  is  the  age  of  each? 

Let  3x  equal  Reuben's  age. 

Then  4x  equals  Silas's  age 

and  5x  equals  Joseph's  age. 

12x  equals  sum  of  their  ages. 

12x  equals  36  years. 

X  equals  3  years. 

3x  equals  9  years,  Reuben's  age. 

4x  equals  12  years,  Silas's  age. 

5x  equals  15  years,  Joseph's  age. 

1.  Gunpowder  is  made  of  2  parts 
sulphur,  1  part  of  saltpetre,  and  3 
parts  charcoal.  How  many  pounds 
of  each  of  other  ingredients  must  be 
used  vjith  250  pounds  of  saltpetre? 

2.  How  many  pounds  each  of 
sulphur,  saltpetre,  and  charcoal  will 
be  required  to  make  1000  pounds  of 
powder  ? 

3.  The  length  of  a  field  is  to  its 
width  as  7  to  5  and  its  perimeter  is 
120  rods.  Find  its  dimensions  and 
area. 

4.  The  length  of  a  field  contain- 
ing 630  acres  is  to  its  width  as  7  to 
4.  Find  the  length  and  width  of  the 
field.     Draw  a  diagram. 

5.  The  length,  breadth  and  thick- 
ness of  a  rectangular  solid  have  the 
ratio  of  5,  4,  and  3,  and  the  sum  of 
its  edges  is  912  inches.  Find  the 
dimensions   of  the  solid. 

6.  The  length,  breadth  and  thick- 
ness of  a  rectangular  solid  are  in 
the  ratio  of  5,  3,  and  2,  and  its  sur- 


face   contains     1550    square    inches. 
Find  its  dimensions, 

7.  The  length,  breadth  and  thick- 
ness of  a  rectangular  solid  have  the 
ratio  of  6,  5,  and  2,  and  its  volume 
is  20580  cubic  inches.  Find  its  area. 
Illustrate. 

8.  John  is  12  years  older  than 
Henry.  In  seven  years  Henry  will 
be  three-fourths  as  old  as  John, 
What  is  the  age  of  each  now? 

9.  William  is  18  years  old  and 
Henry  is  10.  In  how  many  years 
will  Henry's  age  equal  five-sevenths 
of  William's?    Ans.,  10  yr. 

10.  Mr.  Grant  is  52  years  old  and 
his  wife  is  40.  When  they  were 
married  Mrs.  Grant's  age  was  %  of 
Mr.  Grant's.  How  long  have  they 
been  married?    Ans.,  20  yr. 

}i  of  the  time  past  midnight  = 
%  of  the  time  to  noon.  What  is  the 
time? 

Solution . 
^   of  the  time  past  midnight  =  % 
of  time  to  noon. 

^  of  time  past  midnight  =  % 
of  time  to  noon. 

%  of  time  past  midnight  =  % 
of  time  to  noon. 

Ratio  8  to  7. 
Let     8x=time  past  midnight 
Then  7x=time  to  noon 
15x=the  sum 
15x=12  hours 
x=%  of  an  hour 
8x==6%  hours  or  6  hours  and 
24  min.  Ans.,  6:24  a.  m. 

11.  A  pole  76  ft.  long  was  brok- 
en in  such  a  manner  that  ^  of  the 
length  of  one  piece  equals  ^  of  the 
length  of  the  other.  How  long  is 
each  piece?     Ans.,  40  ft.;  36  ft. 

12.  Two-thirds  of  Mary's  age 
equals  }i  of  Homer's  and  the  differ- 
ence in  their  ages  is  2  years.  What 
is  the  age  of  each? 

In  the  first  eight  problems  below 
it  is  assumed  that  A,  B  and  C  own 
equal  shares  of  the  sheep. 


PROPORTIONAL  PARTS 


111 


It  is  best  to  find  the  total  expense 
and  then  find  each  partner's  share  of 
it.  For  example,  in  the  first  prob- 
lem, since  C's  share  of  the  expense 
is  $50,  the  total  expense  is  $150,  and 
the  pasturage  is  therefore  worth 
$1.50  an  acre.  In  partnership  each 
partner  must  pay  all  his  own  pri- 
vate expenses. 

18.  A,  B  and  C  have  a  flock  of 
sheep  of  which  each  owns  a  third. 
A  has  60  acres,  and  B  40  acres  of 
pasture  land.  The  sheep  eat  the 
pasturage  and  C  pays  $50.  Find 
A's  and  B's  share  of  the  money. 
Ans.,  A,  $40;  B,  $10. 

14.  A  has  80  acres,  B  40  acres, 
and  C  30  acres  of  pasture  land  on 
which  the  sheep  graze.  B  and  C 
agree  to  pay  A  $15.  How  much 
should  each  pav  ?  Ans.,  B,  $5 ;  C, 
$10. 

15.  A  has  60  acres,  B  25  acres, 
and  C  15  acres  of  pasture  land  on 
which  the  sheep  graze.  C  pays 
$13.75.  How  much  should  A  and 
B  pay  or  receive?  Ans.,  A  rec. 
$20;  B  pays  $6.25. 

16.  A  has  125  acres  and  B  60 
acres  of  pasture  land  and  they  rent 
from  other  parties  115  acres.  The 
sheep  eat  the  pasturage  which  is 
valued  at  75  cts,  an  acre.  How  much 
should  A,  B  and  C  each  pay  or  re- 
ceive? Ans.,  A  would  receive  $18.- 
75;  B  pay  $30;  C  pay  $75. 

17.  A  has  75  acres  and  B  25 
acres  of  land  on  which  the  sheep  are 
pastured.  C  is  allowed  $15  for  look- 
ing after  the  sheep.  If  the  pastur- 
age is  reckoned  at  60  cts.  an  acre, 
how  much  should  each  pay  or  re- 
ceive? Ans.,  A  rec.  $20;  B  and  C 
each  pay  $10. 

18.  A  has  75  acres  and  B  50 
acres  of  pasture  on  which  the  pastur- 
age is  worth  30  cents  an  acre.  A 
flock  of  sheep,  of  which  A,  B  and  C 
own  equal  shares,  eat  the  pasture.  C 
is  allowed  $20  for  caring  for  the 
sheep.  How  should  the  bills  be  set- 
tled? 


19.  A  has  80  acres  of  land  with 
pasturage  worth  75  cents  an  acre. 
B  has  60  acres  of  land  with  pastur- 
age worth  $1.25  an  acre.  A,  B  and 
C  have  the  same  number  of  sheep 
to  graze  on  the  land.  C  is  allowed 
$15  for  his  work.  How  shall  they 
settle  the  bills?  Ans.,  A  rec.  $10; 
B  rec.  $25 ;  C  pays  $35. 

20.  A  has  40  acres  of  land  worth 
50  cts.  an  acre,  B  has  65  acres  worth 
$1.00  and  C  20  acres  worth  $1.25  an 
acre.  A  and  C  take  care  of  the 
sheep  @  $5  each.  How  should  they 
settle  the  bills?  Ans.,  A  pays  $15; 
B  rec.  $25;  C  pays  $10. 


21.  A  owns  500  sheep,  B  700,  and 
C  800.  A  has  75  acres,  B  100  acres, 
and  C  140  acres  of  land,  the  pastur- 
age of  which  is  valued  at  60  cents 
an  acre.  C  takes  care  of  the  sheep 
for  $15.  If  the  sheep  eat  the  pastur- 
age off  all  the  land,  how  should  the 
bills  be  settled?  Ans.,  A  pays  $6; 
B  pays  $11.40;  C  rec.  $17.40. 

22.  A,  B  and  C  are  partners  in  a 
store  in  the  rates  of  4,  3  and  5.  At 
the  end  of  a  year  they  find  that  the 
gross  gain  is  $5000  and  the  store 
expenses  $1400.  Each  has  spent 
$900  for  private  expenses  during  the 
year.  Find  the  conditoin  of  each 
man's  account.  Ans.,  A  gains  $300; 
C  gains  $600;  B  gains  $0. 

23.  A,  B  and  C  are  partners  in 
the  ratio  of  5,  4  and  3.  Their  gross 
gains  are  $5700.  At  the  end  of  a 
year  their  store  expenses  including 
salary  are  .$2100.  A's  salary  is  $1000 
a  year,  and  their  private  expenses 
are  $850  each.  What  is  the  gain  of 
each  at  the  end  of  a  year?  Ans., 
A  gains  $1650;  B  gains  $350;  C 
gains  $50. 

24.  A  and  B  own  a  store  in  the 
ratio  of  2  to  5,  and  each  is  allowed 
a  salary  of  $800  a  year  for  his  ser- 
vices. The  yearly  gross  gain  is 
$6500,  the  store  expenses  not  includ- 
ing the  salaries  are  $840,  and  each 
spends  $2500  a  year  for  private  ex- 
penses. What  will  be  the  gain  or 
loss  of  each  in  5  years? 


112 


ARITHMETIC 


Ans.,  B  gams  $6000;  A  loses 
$2700. 

25.  A  and  B  own  a  store  in  the 
ratio  of  5  to  2.  A  is  allowed  $1500 
and  B  $1000  for  personal  services. 
The  gross  gain  is  $6500.  The  store 
expenses,  including  salaries,  is  $3,- 
350.  A's  private  expenses  are  $2500 
and  B's  are  $2000.  Find  the  gain  or 
loss  of  each.  Ans,,  B  loses  $100; 
A  gains  $1250. 

26.  A  machine  is  sold  for  $100, 
one-fifth  of  the  sales  being  profit. 
In  manufacturing  the  machine  the 
cost  of  labor  is  to  the  cost  of  the  ma- 
terial as  7  is  to  9,  If  the  cost  of  la- 
bor advances  one-fifth  and  that  of 
material  falls  one-fifth,  and  the  ma- 
chine sells  for  $100  what  vvfill  the 
profit  be?    Ans.,  $22  gain. 

27.  Mesdames  Brown,  Jones  and 
Smith  prepared  a  dinner.  Mrs. 
Brown  furnished  fruit  and  vegeta- 
bles valued  at  $3,  Mrs.  Jones  fur- 
nished meat  and  pastries  valued  at 
$4.50.  Bread  was  purchased  and 
dishes  rented  at  an  expense  of  $2.50. 
Mrs.  Smith  did  the  work  for  $5. 
The  Brown  family  numbered  9 ;  the 
Jones  family  10,  and  the  Smith  fami- 
ly 11  How  should  the  bills  be  set- 
tled? Ans.,  Mrs.  J.  pay  $.50;  Mrs. 
S.   pay  $.50;  Mrs.    B.   pay  $1.50. 

28.  Fast,  Slow  and  Steady  owned 
a  dairy  in  the  ratio  of  7,  6,  and  9. 
Fast  received  $30  a  month  wages ; 
Slow,  $18;  and  Steady,  $25.  The 
feed  and  other  expenses  cost  $120. 
The  receipts  for  the  month  were 
$270.  Find  each  man's  share.  Ans., 
Fast,  $54.50;  Slow,  $39.00;  Steady, 
$56.50. 

29.  A,  B  and  C  are  partners  in 
farming.  A  furnishes  250  acres  of 
land  which  rents  at  $3  an  acre,  B 
furnishes  seed  and  provisions  worth 
$500,  and  C  furnishes  stock,  hay  and 
implements,  the  use  of  which  is  val- 
ued at  $450.  Each  is  allowed  $200 
a  year  for  wages.  The  crop  sells 
for  $3140,  and  the  personal  expens- 
es are  $450,  $350,  and  $300  respec- 
tively.   Find  the  gain  or  loss  of  each. 


the  profits  being  shared  equally.  How 
much  money  will  each  have  at  the 
end  of  the  year?  Ans.,  A  has  $780; 
B,  $630;  C,  $630. 

Proportion. 

Simple  proportion  consists  of  two 
pairs  of  equal  ratios.  Thus,  %  equal 
"^%7>  the  ratio  27  yards  to  18  yards 
equals  the  ratio  of  $9  to  $6  for  -y^s 
equals  %. 

A  proportion  may  be  expressed  as 
follows:  i%o  =  2%5,  16:10  =  24:15, 
or  16:10  ::  24:15.  The  first  form 
given  is  preferable. 

The  first  and  fourth  terms  of  a 
proportion  (16  and  15  in  the  above) 
are  called  the  extremes  and  the  sec- 
ond and  third  (10  and  24)  the 
means. 

In  every  proportion  the  product  of 
the  extremes  equals  the  product  of 
the  means. 

One  quantity  varies  as  another 
when  an  increase  or  decrease  in  one 
causes  a  corresponding  increase  or 
decrease  in  the  other,  that  is  when 
if  one  is  multiplied  or  divided  by  a 
number  the  other  is  multiplied  or  di- 
vided by  the  same  number.  The 
distance  one  can  travel  varies  as  the 
time  when  the  rate  is  the  same.  The 
amount  of  grain  produced  will  vary 
as  the  amount  of  land  if  the  produc- 
tion per  acre  is  the  same.  It  will 
vary  as  the  amount  produced  on  one 
acre  on  the  same  number  of  acres. 

One  quantity  varies  inversely  as 
another  when  if  one  is  multiplied  or 
divided  by  any  number  the  other  is 
divided  or  multiplied  by  the  same 
number.  The  time  required  to  trav- 
el a  given  distance  varies  inversely 
as  the  rate.  The  number  of  articles 
that  can  be  bought  with  a  given 
amount  of  money  varies  inversely  as^ 
the  price  of  one  article.  The  num- 
ber of  men  required  to  do  a  piece  of 
work  varies  directly  as  the  amount 
of  work  and  inversely  as  the  time  in 
which   it  must  be  done. 

If  a  train  travels  486  miles  in  14 
hours,  how  many  hours  will  it  take 
to  travel  3000  miles  at  the  same  rate  ? 


PROPORTION 


113 


Solution. 

hours.  miles. 

14  486 

?  3000 

Let  X  equals  time  required  to  trav- 
el 3000  mi. 

Then  x'/i4  =  sooo/^g^ 
X  =  14X3000 


486 
86.4  plus  hr.     Ans. 

If  75  men  can  do  a  piece  of  work 

in    30   days,    how   many   men   would 

be  required  to  do  it  in  18  days? 

men.  days. 

75  30 

?  18 

Let    X    equal    number    of    men    to 
do  the  work  in  18  da. 

x/t5  =  -^ris 

X  =  75  X  s%8  equals  125. 
125  men  Ans. 

The    work    may    be    shortened    by 
omitting  the  first  equation. 

If  3800   sacks   of  wheat  are   pro- 
duced   on    320    acres    of    land,    how 
many  acres  of  land  of  the  same  grade 
are  required  to  produce  12000  sacks? 
sacks.  acres. 

3800  320 

12000  ? 

Let  X  equal  acres  required  to  pro- 
duce 12000  sacks. 

Then  x  =     320  A.  X 12000 


3800 


X  =  1010.52  A. 


These  solutions  suggest  the  rule. 

RULE:  Place  the  unknown  term 
equal  to  the  like  known  multi- 
plied by  the  ratio   of  the   other  two, 

that  ratio  being  made  greater  than 
one  when  the  answer  should  be 
greater,  and  less  than  one  when  it 
should  be  less  than  the  term  like  the 
required  answer. 

1.  If  37  acres  of  prunes  bring  a 
net  income  of  $3500,  what  should  be 


the  net  income  of  a  similar  crop  on 
75   acres. 

2.  If  land  which  nets  its  owner 
$9  an  acre  yearly  is  valued  at  $75 
an  acre,  what  is  the  value  of  land 
which  nets  $200  an  acre  yearly? 

3.  If  a  train  makes  a  certain  dis- 
tance in  27  hours  traveling  35  miles 
an  hour,  how  long  will  it  take  an 
aeroplane  traveling  76  mi.  an  hour 
to  make  the  same  distance? 

4.  Twenty- five  tons  of  ore  of  a 
certain  mine  produce  metal  worth 
$417.65,  how  much  of  the  same  ore 
is  required  to  yield  metal  worth 
$4000  ? 

5.  A  contractor  engaged  to  com- 
plete a  piece  of  road  work  in  30 
days.  He  employed  26  m.en,  and 
found  at  the  close  of  18  days  that 
Yz  the  work  was  done.  How  many 
additional  men  must  be  employed 
that  the  work  may  be  completed  in 
the  required  time? 

6.  A  ship  started  on  a  cruise  of 
90  days  with  provisions  for  its  125 
men.  At  the  end  of  70  days  25  men 
left  the  ship.  How  long  will  the 
provisions   last  the   remainder? 

Compound  Proportion. 

Sometimes  one  ratio  depends  up- 
on the  combined  effect  of  two  or 
more  ratios.  The  distance  traveled 
depends  on  both  the  rate  and  the 
time.  The  value  of  a  crop  depends 
upon  both  the  price  and  the  amount 
produced.  Ratios  thus  combined 
constitute  a  compound  ratio  and  the 
resulting  proportion  a  compound  pro- 
portion. 

The  solution  of  a  compound  pro- 
portion does  not  differ  materialy 
from  that  of  a  simple  proportion. 

Write  the  unknown  term  equal  to 
the  like  known  multiplied  by  the  ra- 
tios of  the  other  like  terms.  The 
terms  of  each  ratio  must  be  arranged 
as  if  the  result  depended  on  it  alone. 

If  it  costs  $85  to  floor  a  room 
14'  by  24',  how  much  will  it  cost  to 


114 


ARITHMETIC 


floor  a  room  16'  by  22'  with  flooring 
of  the  same  price? 

cost.               width.  length. 

$85                 14  24 

?                  16  22 

Let     X  =  cost   of  flooring    room 
16'  by  22'. 

Thenx  =  $85Xi%4X22/24 

X  =  $89.10    Ans. 
If  200  men  construct  18  miles  of 
road   in   55   days,   how   long   will    it 
take  220  men  to  construct  27  miles? 
men.  days.  miles. 

200  55  18 

220  ?  27 

Let     X  =  days   it  takes   220  men 
to  construct  27  miles. 

Thenx  =  55  days  X  200/220X2-/18 
X  ==  75  days    Ans. 

220  men  can  construct  the  road  in 
less  time  than  it  takes  200  men, 
hence  the  ratio,  -o%2o,  must  be  less 
than  1. 

It  will  take  longer  to  construct 
27  miles  than  18  miles,  hence  the 
ratio,  -%8>  must  be  greater  than  1. 

1.  If  the  peach  crop  from  25 
acres  brings  $3400  when  the  dried 
fruit  is  sold  at  7c  a  pound,  what  is 
the  value  of  a  similar  crop  on  45 
acres  sold  at  6c  a  pound? 

2.  If  4200  tons  of  iron  ore  pro- 
duce $3430  when  iron  is  $36.80  a 
ton,  what  is  the  value  of  3500  tons 
of  like  ore  when  iron  is  $42.50  a 
ton? 

3.  If  a  reservoir  600  feet  long, 
350  feet  wide  and  9  feet  deep  holds 
450000  barrels,  how  much  will  a 
reservoir  800  feet  long,"  375  feet 
wide  and  7  feet  deep  hold? 

Similar  Figures. 

Similar  surfaces  and  solids  are 
those  which  have  the  same  shape. 
Their  corresponding  angles  must  be 
equal  and  their  corresponding  lines 
are  proportional. 

It  is  true  of  similar  figures  that 
(1)  their  corresponding  lines  are 
proportional,   (2)   the  areas     of  sim- 


ilar are  to  each  other  as  the  squares 
of  their  corresponding  hnes,  (3)  the 
volumes  of  similar  figures  are  to 
each  other  as  the  cubes  of  their 
corresponding  lines. 

The  dimensions  of  a  block  of  mar- 
ble are  8  in.,  6  in.,  and  4  in.,  the  cost 
of  polishing  it  $.65,  and  its  weight  is 
18.26  lb.  The  length  of  a  similar 
block  is  42  in.  Find  the  width, 
thickness,  and  weight  of  the  second 
block  and  the  cost  of  polishing  its 
surface. 

1.       w.      th.     cost       wt. 
8"     6"     4"     $.65     18.26 
42        ?        ?         ?  ? 

w.   equals   6"  X  42, 

th.  equals  4"  X  42, 

cost  equals  $ .  65  X  42^, 

~82 
wt.   equals   18.26   lb.  X  42^. 

1.  A  cone  is  6  inches  in  diameter 
and  7  inches  in  altitude.  What  is 
the  height  of  a  similar  cone  11  inch- 
es in  diameter? 

2.  If  it  takes  24  thousand  shingles 
to  cover  a  roof  40  by  65  feet,  how 
many  will  it  take  to  cover  both  sides 
of  a  similar  roof  30  feet  wide? 

3.  If  a  man  6  feet  tall  weighs 
180  lb.  what  did  Goliath  who  was 
10.5  feet  tall  weigh  if  he  was  of 
similar    proportions  ? 

4.  If  a  6  inch  cannon  ball  weighs 
39.5  pounds  what  is  the  weight  of  a 
14  inch  ball  of  the  same  material? 

Longitude  and  Time. 

The  earth  revolves  on  its  axis  from 
west  to  east,  causing  the  succession 
of  day  and  night.  It  makes  a  com- 
plete revoliTtion  in  24  hours ;  hence, 
at  a  given  place  it  is  24  hours  from 
sunrise  to  sunrise,  or  from  noon  to 
noon. 

Show  that  the  following  statements 
are  true : 


LONGITUDE  AND  TIME 


115 


The  earth  turns  360°  in  24  hr. 


15°  " 

1 

hr. 

1  O     " 

4 

min 

15'  " 

1 

min. 

r  " 

4 

sec. 

15"  " 

1 

(< 

Stand  with  your  face  to  the  south 
and  point  toward  the  place  where  the 
sun  is  seen  at  noon.  Will  it  be  fore- 
noon or  afternoon  with  persons  living 
east  of  you?  West  of  you?  It  is 
noon  now  at  some  place.  Is  that 
place  east  or  west  of  here?  Where 
was  it  noon  an  hour  ago?  Where 
will  it  be  noon  an  hour  hence? 

The  time  of  day  is  later  east  and 
earlier  west  of  a  given  place. 

Examine  the  following  statements 
and  tell  why  they  are  true : — 

A  difference  of  15°  in  Ion.  makes 
a  difference  of  1  hour  in  time. 

A  difference  of  15'  in  Ion.  makes 
a  difference  of  1  min.  in  time. 

A  difference  of  15"  in  Ion.  makes 
a  difference  of  1  sec.  in  time. 

Difference  in  longitude  may  be 
changed  to  difference  in  time  by  di- 
viding by  15  and  calling  the  quotient 
hrs.,  min.  or  sec,  according  as  the 
dividend  is  degrees,  minutes,  or  sec- 
onds. 

Difference  in  time  may  be  changed 
to  difference  in  longitude  by  multi- 
plying by  15  and  calling  the  result 
degrees,  minutes,  or  seconds  accord- 
ing as  the  multiplicand  is  hours, 
minutes,  or  seconds. 

Thru  the  relation  of  longitude  and 
time  a  captain  determines  the  longi- 
tude of  his  ship  at  sea,  and  the  lon- 
gitude of  a  place  on  the  land  is 
found. 

1.  Find  the  difference  in  time  be- 
tween (1)  3:30  a.  m.  and  7:15  a.  m. ; 
(2)  6:25  a.  m.  and  4:45  p.  m. ;  (3) 
1:10  p.  m.  and  11:55  p.  m. ;  (4) 
2:55  a.  m.  and  9:12  p.  m. ;  (5)  7:18 
p.  m.  and  4'27  a.  m.  next  day;  (6) 
8:20  a.  m.  and  10:40  p.  m.  next  day; 
(7)  11:27  a.  m.  and  6:45  a.  m.  next 
day;  (8)  6:15  a.  m.  and  4:35  p.  m. 
next  day;  (9)  3:30  p.  m.  and  8:15 
a.    m.     preceding    day;     (10)     9:27 


p.  m.  and  5:15  a.  m.  preceding 
day;  (11)  2:55  p.  m.  and  11:25 
p.   m.   of  preceding  day. 

2.  What  is  the  time  9  hr.  24  min. 
later  than  the  following  times:  (1) 
2:30  a.  m.  ?  (2)  7:45  a.  m.? 
(3)   1:12  p.   m.     (4)   11:38  p.   m.? 

Find  the  time  which  is  15  hr.  40 
min.  earlier  than  the  following 
times:  (1)  6:15  a.  m. ;  (2)  1:50  a. 
m.;  (3)  9:10  p.  m.  ;  (4)  2:15  p.m. 

Longitude  is  reckoned  east  or 
west  from  an  established  meredian. 
The  Meridian  of  Greenwich  is  used 
in  the  problems  in  this  book,  that 
being  the  one  commonly  so  used. 


«•'     if  fco*  ^i"  Jo* 


,r* 


f^  ^°'  Hi-"  (^'  ',•>■'  <J3' 


^v 


4.  Find  the  difference  in  longi- 
tude between  the  following  places: 
(1)  15°  W.  and  75°  W.  ;  (2) 
15°  E.  and  75°  W.  ;  (3) 
30°  E.  and  45°  E.  :  (4)  30°  E.  and 
45°  W. 

5.     Find  the  longitude  of  a  person 
who  has  traveled  as   follows: 


(1)  Started  at  45' 
60°  eastward; 


E.  and  traveled 


116 


ARITHMETIC 


(2)  Started  at  45°  E.  and  traveled 
70°  westward; 

(3)  Started  at  23°  45'  W.  and 
traveled  65°  eastward; 

(4)  Started  at  35°  W.  and  trav- 
eled 20°  30'  eastward; 

(5)  Started  at  26°  27'  W.  and 
traveled  38°  50'  westward. 

Places  on  the  earth  cannot  have 
more  than  180°  E.  or  W.  longitude. 

6.  The  longitude  of  San  Francisco 
is  122°  W.,  nearly.  Find  the  longi- 
tude of  a  person  who  starts  at  San 
Francisco  (1)  30"^  W. ;  (2)  65°  E., 
(3)  150°  E.  :  (4)  75°  E. 

7.  If  one  should  start  at  Calcutta 
(88°  E.)  and  travel  125°  eastward, 
what  would  then  be  his  longitude? 

The  time  of  day  is  determined  by 
the  revolution  of  the  earth  on  its 
axis,  the  date  and  the  day  of  the 
week  are  fixed  by  man.  In  order  to 
avoid  confusion  of  date  the  nations 
have  agreed  upon  an  International 
Date  Line.  This  line  runs  along 
180°  E.  or  W.  except  where  the 
latter  runs  across  land,  or  would 
separate  islands  of  a  group.  In  such 
cases  the  date  line  runs  east  or  west 
of  180°.  The  new  day  first  begins 
at  midnight  on  the  International 
Date  Line. 

The  time  of  any  place  in  E.  lon- 
gitude is  always  later  than  that  of 
any  place  in  W.    longitude. 

When  the  difference  in  longitude 
in  two  places  is  found  to  be  greater 
than  180°,  it  is  not  best  to  subtract 
that  amount  from  360°  before  find- 
ing the  difference  in  time. 

To  avoid  confusion  in  time  tables 
and  guard  against  accidents,  the 
railway  managers  of  North  America 
have  agreed  on  a  system  of  time- 
keeping, called  Standard  Time. 
Meridians  are  chosen  fifteen  degrees 
apart,  and  all  places  near  one  of 
these  meridians  keep  the  time  of  that 
one.  It  follows  that  places  near 
different  meridians  will  keep  times 
differing  by  one  or  more  hours.  The 
minute  hands  of  all  clocks  keeping 
Standard  Time  will  agree. 


Eastern  Standard  Time  is  the  time 
of  75°  W.;  Central  Standard  Time, 
the  time  of  90°  W. ;  Mountain  Time, 
of  105°  W.;  Pacific  Standard  Time, 
of  120°  W. 

8.  How  much  does  each  of  these 
times  differ  from  the  time  of  Green- 
wich? 

9.  Washington  is  77°  W.  How 
much  does  the  Stadard  Time  differ 
from  its  local  time? 

10.  What  is  the  difference  be- 
tween Eastern  Time  and  Pacific 
Time? 

11.  Boston  is  71°  W.  Is  Stand- 
ard Time  too  fast  or  too  slow  for 
Boston,  and  how  much? 

12.  When  it  is  1  hr.  20  min.  p.m. 
Saturday,  at  110°  E.,  at  what  place 
is  9  p.  m.  ?  What  is  the  day  at  the 
latter  place?    Ans.,  Friday,  135°  W. 

13.  What  is  the  date  and  hour  at 
S.  F.,  122°  W.,  when  it  is  8  a.  m., 
May  1st,  at  Manila,  121°  E.?  Ans., 
3:48  p.   m.,  April  30. 

14.  What  is  the  date  and  hour  at 
Pekin,  116°  E.,  when  it  is  11:40 
a.  m.,  Oct.,  20th,  at  S.  F.,  122°  W.? 
Ans.,  3:32  a.  m.,  Oct.  21. 

15.  What  is  the  time  at  Calcutta, 
88°  E.,  when  it  is  2  hr.  25  min.  p.  m. 
at  St.  Paul,  93°  W.  ?  Ans.,  2:2J 
a.   m.,  next  day. 

16.  A  ship  which  carries  S.  F. 
(122°  W.)  time,  finds  the  local  time 
to  be  3  hr.  45  m.  p.  m.,  when  the 
S.  F.  time  is  10  hr.  20  m.  a.  m. 
What  is  the  longitude  of  the  ship? 
Ans.,  40°  45'  W. 

17.  At  another  time  the  local  time 
is  found  to  be  8  hr.  27  m.  a.  m. 
when  the  S.  F.  time  is  2  hr.  15 
m.  p.  m.  Where  is  the  ship?  Ans., 
151°  E. 

18.  Where  is  the  ship  when  the 
local  time  is  4  hr.  45  m.  p.  m., 
when  the  S.  F.  time  is  6  hr.  30  m. 
a.   m.?    Ans.,  31°  45'  E. 


PROBLEMS 


117 


19.  If  a  ship  carries  Greenwich 
time,  what  is  its  longitude  when  the 
local  time  is  5  hr.  35  m.  p.  ir.  and 
the  Greenwich  time  is  2  hr.  50  m. 
a.  m.?  If  it  is  Monday  at  Green- 
wich, what  is  the  day  where  the 
ship  is  situated?  Ans.,  138="  45'  W. ; 
Sunday . 

20.  When  it  is  9:00  p.  m.  on 
Friday  at  110°  W.  it  is  2:20  p.  m. 
at  another  place.  What  is  the  longi- 
tude of  the  latter  place?  What  is 
the  day  of  the  week?  Ans.,  150°  E., 
Saturday. 

21.  When  it  is  2:20  p.  m.  Sun- 
day at  120°  E.  longitude,  it  is  8:05 
p.  m.  at  another  place.  What  is  the 
longitude  of  the  latter  place,  and 
what  day  of  the  week  is  it?  Ans., 
153°  45'  W.;  Saturday. 

22.  The  exact  local  time  at  Lick 
Observatory  is  8  hr.  6  min.  31.85 
sec.  slower  than  Greenwich  time. 
What  is  the  longitude  of  the  Obser- 
vatory?    Ans.,  121°  37'  57.75"  W. 

23.  When  it  is  1 :35  p.  m.  Saturday 
at  110°  E.,  what  is  the  hour  and  day 
at  110°  W.?  Ans.,  10:55  p.  m. 
Friday. 

24.  A  watch  is  set  right  at  Pekin, 
116°  30'  E.  On  what  meridian  is 
the  watch  when  at  noon  it  shows 
9:20  a.  m.?    Ans.,  156°  30'  E. 

25.  When  it  is  5  minutes  after  4 
o'clock  on  Sunday  morning  at  Hono- 
lulu, longitude  157°  52'  W.,  what  is 
the  time  at  Sidney,  Australia,  longi- 
tude 151°  11'  E.  Ans.,  41  m.  12  sec. 
a.  m. 

26.  When  it  is  10  o'clock  p.  m. 
Wednesday,  at  longitude  20°  E., 
what  is  the  time  at  San  Francisco, 
122°  W.?    Ans.,  12:32  p.  m. 

27.  The  time  which  is  telegraphed 
over  the  state  from  Mt.  Hamilton  is 
Pacific  Standard  Time.  What  is  the 
difference  between  this  and  the  ex- 
act time  of  the  Observatory? 

Sawing  Wood,  Cutting  Ice,  Etc. 
It   is    supposed    in    these   problenis 


that   the   charge   is   in   proportion  to 
the  amount  of  work  done. 

1.  If  it  costs  $12  to  survey  the 
sides  of  a  square  forty  acre  field, 
how  much  will  it  cost  to  divide  it 
into  square  lots  of  two  and  a  half 
acres  each?     Ans.,  $18. 

2.  It  cost  $510  to  fence  a  field 
50  rods  by  70  rods.  How  much  ad- 
ditional will  it  cost  to  divide  the  field 
into  lots  10  rods  square?  Ans., 
$1232.50. 

3.  If  it  costs  $336  to  fence  a 
field  36  rods  by  60  rods,  how  much 
additional  will  it  cost  to  fence  it 
when  divided  into  the  largest  possi- 
ble equal  square  lots?     Ans,,  $462. 

4.  It  cost  $1050  to  build  a  solid 
board  fence  about  a  lot  600  feet  by 
800  feet.  How  much  additional  should 
be  paid  to  fence  it  into  lots  200  feet 
by  150  feet?     Ans.,  $1575. 

5.  If  60  cts.  a  cord  is  charged  for 
sawing  4  ft.  wood  into  16  in.  sticks, 
how  much  should  be  charged  for 
sawing  the  same  wood  into  12  in. 
sticks?     Ans.,  90  cts. 

The  expense  should  be  in  propor- 
tion to  the  amount  of  sawing,  and 
not  to  the  number  of  sticks  into  which 
each  stick  of  cordwood  is  cut.  A 
diagram  will  be  found  helpful.  For 
problems  like  the  sixth,  note  that  a 
stick  of  eight  foot  wood  contains 
twice  as  much  wood  as  a  stick  of 
four  foot  wood. 

6.  If  75  cts.  a  cord  is  charged 
for  sawing  4  ft.  wood  into  12  in. 
sticks,  how  much  a  cord  should  be 
charged  for  sawing  8  ft.  wood  into 
16  in.  sticks?     Ans..  62i/^  cts. 

7.  75  cts.  a  cord  is  charged  for 
sawing  4  ft.  wood  into  12  in.  sticks, 
the  charge  including  25  cts.  for  a 
helper.  If  30  cts.  a  cord  is  allowed 
for  a  helper,  how  much  should  be 
charged  for  sawing  8  ft.  wood  into 
16  in.  sticks?     Ans.,  71^  cts. 

8.  If  50  cts.  a  cord  is  charged  for 
sawing  2  ft.  wood  into  sticks  12  ins. 
long,   how   much    should   be   charged 


118 


ARITHMETIC 


for  sawing  5  cords  of  4  ft.  wood  into 
sticks  of  the  same  length?  Ans., 
$3.75. 

9.  If  7  cts.  is  paid  for  sawing  a 
block  of  ice,  9X10X1  ft.,  into  pieces 
2X3X1  ft.,  what  should  be  charged 
for  sawing  a  block  9X12X1  ft.  into 
pieces  3X4X1  ft?     Ans.,  5}i  cts. 

10.  If  6  cts.  is  charged  for  saw- 
ing a  block  of  ice  9X10X1  ft.  into 
pieces  3X5X1  ft.,  how  much  should 
be  charged  for  sawing  a  block  12X 
15X1  ft.  into  pieces  3X6X1  ft.? 
Ans.,  13^9  cts. 

11.  If  11  cts.  is  charged  for  saw- 
ing a  block  of  ice  20'X30'X2'  into 
pieces  4'X5'X2' ,  how  much  should 
be  charged  for  sawing  a  block  15' 
24' X  3'    into  pieces  3'X6'X3'  ? 

Ans.,  102%o  cts. 

12.  If  24  cts.  is  charged  for  paint- 
ing a  cubic  foot  of  wood  on  the  out- 
side, how  much  additional  should  be 
charged  for  painting  it  when  divided 
into  pieces  2X3X4  in.? 

Ans.,  80  cts. 

13.  $13.05  is  paid  for  polishing  a 
block  of  stone  8X10X12  ft.  How 
much  additional  should  be  paid  for 
polishing  the  same  when  divided  into 
pieces  3X4X5  ft.?     Ans.,   $20.10. 

14.  If  it  takes  72  hours  to  lath  a 
room  48X60X15  ft.,  how  long  will 
it  take  when  it  is  divided  into  rooms 
12X16X15  ft?     Ans.,  I822/17  hrs. 

Working  Problems. 

When  money  is  to  be  paid,  divide 
it  among  the  workers  in  proportion 
to  the  amount  of  work  done. 

1.  A  and  B  together  can  do  a 
piece  of  work  in  15  days.  After 
working  together  6  days,  A  leaves 
and  B  finishes  the  work  in  30  days 
more.  In  how  many  days  can  each 
alone  do  the  work? 

A,  2134  da.;  B,  50  da. 

2.  A  and  B  together  can  do  a 
piece  of  work  in  12  days.  After 
working  together  for  9  days,  how- 
ever, they  call  in  C  to  help  them,  and 


the  three  finish  the  work  in  2  days. 
In  how  many  days  can  C  alone  do 
the  work?  Ans.,  24  da. 

3.  Henry  can  do  a  certain  piece 
of  work  in  18  hours,  John  can  do 
the  same  in  12  hours,  and  their  fath- 
er in  6  hours.  Henry  begins  work 
at  7  o'clock,  John  begins  at  8,  and 
their  father  is  to  begin  in  time  for 
the  work  to  be  finished  by  noon. 
When  must  their  father  begin  work? 

Ans.,   9:40. 

4.  A  can  do  a  piece  of  vv^ork  in 
20  days,  A  and  B  in  12  days,  B  and 
C  in  10  days.  In  what  time  can  C 
do  the  work  alone?       Ans.,  15  da. 

5.  Three  men,  4  women,  or  5 
boys  can  do  a  job  of  work  in  8 
hours ;  in  what  time  can  1  man,  2 
women,  and  3  boys  do  it  working 
together?  Ans.,   0^%^   hr. 

6.  One  pipe  can  fill  a  tank  in  8 
hours,  a  second  in  11  hours,  a  third 
can  empty  it  in  15  hours.  If  the 
tank  is  empty  and  the  pipes  are  all 
opened,  in  what  time  will  the  tank 
be  filled?  Ans.,  6^'^%qi  hr. 

7.  A,  B  and  C  can  do  a  job  of 
work  in  10,  12,  and  15  days  respec- 
tively. A  works  4  days,  B  3  days, 
and  C  finshes  the  work.  If  $30  is 
paid  for  the  work,  how  much  should 
each  receive? 

Ans.,  A,  $12 ;  B,  $7.50 ;  C,  $10.50. 

8.  A,  B  and  C  together  can  do  a 
piece  of  work  in  10  days ;  A  and  B 
together  in  12  days;  B  and  C  to- 
gether in  20  days.  How  long  will 
it  take  each  alone  to  do  the  work? 

9.  Henry  and  Samuel  could  have 
done  a  piece  of  work  in  15  hours, 
but  after  working  together  for  6 
hours,  Samuel  was  left  to  finish  it, 
which  he  did  in  30  hours.  In  what 
time  could  Henry  have  finished  the 
work  if  Samuel  had  left  at  the  end 
of  6  hours? 

10.  A  can  dig  a  well  in  8  days, 
and  B  in  12  days.  They  work  at  it 
on  alternating  days,  A  beginning. 
How  long  will  it  take  to  dig  the 
well?     If  $24  is  paid  for  digging  the 


PROBLEMS 


119 


well,  how  much  should  each  re- 
ceive ? 

Ans.,  9^2  days;  A  receives  $15; 
B,  $9. 

,  It  is  necessary  in  problems  like 
the  tenth  to  find  the  period  which  is 
repeated  and  deal  with  it  as  a  whole. 
A  does  one-eighth  of  the  work  in 
one  day,  and  B  one-twelfth,  hence 
they  do  together  five-twenty  fourths 
in  two  days.  Since  they  work  on  al- 
ternate days,  the  repeating  period  is 
two  days.  It  will  take  four  whole 
periods  and  there  will  remain  four- 
twentyfourths.  On  the  day  follow- 
ing the  fourth  period  A  does  three- 
twentyfourths,  leaving  one-twenty- 
fourth  for  B  to  finish.  He  can  do 
this  in  one-half  a  day. 

11.  A  can  do  a  job  in  10  days, 
B  in  12  days.  If  A  works  in  the 
afternoons  only,  and  B  works  all 
day,  how  long  will  it  take  to  do  the 
work  ?  How  much  should  each  re- 
ceive  if  $15   is   paid    for   the   work? 

Ans.,  7%i  da. :  A  rec.  $5%i ;  B 
rec.  $96/ii. 

12.  A  can  do  a  piece  of  work  in 
10  days,  B  in  12  days,  A  and  B 
work  on  alternate  forenoons,  A  be- 
ginning: both  work  in  the  after- 
noon. How  long  will  it  take  to  do 
the  work?  $18  is  paid  for  the 
work.  How  much  should  each  re- 
ceive? Ans.,  7%  da, 

13.  A  man  and  a  boy  undertake 
a  piece  of  work.  The  man  alone  can 
do  the  work  in  8  days,  and  the  boy 
in  12  If  the  man  begins  work  on 
the  first  day  and  works  every  other 
day  only,  and  the  boy  works  every 
day  from  the  first,  how  long  will  it 
take  them  to  complete  the  work? 

Ans.,  6^  da. 

14.  A  can  do  a  piece  of  work  in 
10  hours  and  B  in  12.  They  begin 
the  work  together  at  7  a.  m.  A 
works  an  hour  and  rests  half  an 
hour,  and  so  continues.  B  works 
all  the  time.  When  will  the  work 
be  completed?     Ans.,   1:35 — p.  m. 

Traveling   and   Rowing. 
1.     Train  No.  1,  which  travels  24 


miles  per  hour,  passes  a  stake  in  15 
sec.  Train  No.  2,  which  travels  30 
miles  per  hour,  passes  a  stake  in  16 
sec.  How  long  is  each  train?  How 
long  will  it  take  the  trains  to  pass 
each  other  on  a  double  track,  if 
going  in  opposite  directions?  How 
long  if  going  in  the  same  direction? 
Ans.,  length  of  trains,  528  ft., 
704  ft. 

2.  It  is  120  miles  from  Yolo  to 
Milpitas.  Train  No.  1  (above)  leaves 
Yolo  at  7  a.  m.  to  go  to  Milpitas 
and  return,  and  stops  15  min.  at 
Milpitas.  Train  No.  2  (above) 
leaves  Yolo  for  Milpitas  at  9  a.  m. 
Where  will  trains  meet? 

Ans.,   10  miles   from   Milpitas. 

Find  the  location  of  Train  No.  2 
when  Train  No.  1  starts  back  from 
Milpitas.  The  first  train  reaches 
Milpitas  at  12  m.  and  starts  back  at 
12 :15.  The  second  train  will  be 
97>4  miles  from  Yolo  and  22>^  miles 
from  Milpitas  at  that  time.  They 
now  approach  each  other  at  a  com- 
bined rate  of  54  miles  an  hour  and 
will  meet  in  25  minutes  10  miles 
from  Milpitas. 

In  problems  where  there  is  a 
change  of  rate  or  a  stop  find  the 
position  of  the  parties  after  the  last 
stop  or  change  and  then  there  will 
be  little  difficulty  in  completing  the 
work. 

3.  The  distance  from  San  Jose 
to  San  Francisco  is  51  miles.  A 
can  ride  the  distance  in  4^4  hours, 
B  in  5%o  hours.  A  leaves  S.  F.  for 
San  Jose  at  8  o'clock ;  B  leaves  S. 
J.  for  S.  F.  at  9  o'clock.  When 
will  they  meet? 

Ans.,   10:46   o'clock. 

4.  A  and  B  start  at  7  o'clock  a, 
m.  to  travel  over  the  mountains  into 
San  Joaquin  valley.  A  travels  6 
miles  an  hour  up  hill  and  9  miles  an 
hour  down  hill.  B  travels  5  miles  an 
hour  up  hill  and  10  miles  an  hour 
down  hill.  It  is  20  miles  to  the  top 
of  the  gfrade.  When  and  where  will 
they  next  be  together? 

Ans.,  5  p.  m. ;  80  miles  from  San 
lose. 


ARITHMETIC 


5.  If  A  can  go  from  Albany  to 
Boston  in  9%  hours,  and  B  from 
Boston  to  Albany  in  ll^^  hours,  and 
they  start  at  the  same  time,  in  how 
many  hours  will  they  meet? 

Ans.,  521425  hrs. 

6.  A  train  600  feet  long  is  trav- 
eling 25  miles  an  hour.  How  long 
will  it  take  it  to  pass  entirely 
through    a    tunnel    1800    feet    long? 

Ans.,   lYii   min. 

7.  James  starts  from  San  Fran- 
cisco, which  is  50  miles  away,  at  8 
a.  m.  He  rides  8  miles  an  hour  for 
3  hours,  is  delayed  one  hour  by  an 
accident,  and  proceeds  at  6  miles  an 
hour.  Silas  starts  at  9  a.  m.  and 
rides  7  miles  an  hour.  Where  will 
he  pass  James? 

Ans.,  42  miles   from   S.   F. 

8.  A  wagon  leaves  San  Jose  at 
7  a.  m.  to  go  to  the  coast,  traveling 
5  miles  an  hour  up  hill  and  8  miles 
an  hour  down  hill.  A  carriage 
leaves  at  8  a.  m.  and  travels  6 
miles  an  hour  up  hill  and  10  miles 
an  hour  down  hill.  It  is  20  miles 
to  the  top  of  the  grade  and  40  miles 
from  the  top  to  the  point  on  the 
coast.  Where  will  the  carriage  over- 
take the  wagon? 

Ans.,  33^  miles  from  San  Jose. 

9.  A  man  can  row  5  miles  an 
hour  in  still  water.  How  fast  can 
he  row  against  a  current  which  runs 
3  miles  an  hour?  How  fast  can  he 
row   with   the   same   current? 

10.  How  long  will  it  take  the 
man  to  row  up  stream  12  miles  and 
back? 

11.  How  far  can  he  row  down 
stream  and  back  in  5  hours? 

12.  A  team  can  travel  up  hill  3 
miles  an  hour  and  down  hill  8  miles 
an  hour.  How  far  can  the  team 
travel  up  hill  and  back  in  7  hours? 

13.  A  man  who  travels  by  team 
10  miles  an  hour,  and  on  foot  4 
miles  an  hour,  has  a  journey  of  36 
miles  to  make.  How  far  must  he 
!go   by   team    that   he   may   complete 


the   journey   in   six   hours    from   the 
time  of  starting?         Ans,,  20  mi. 

14.  A  and  B  started  from  M  and 
N  respectively  and  traveled  till  they 
met,  when  it  was  learned  that  A 
had  traveled  five-sevenths  as  far  as 
B.  If  A  had  traveled  18  miles  far- 
ther, and  B  had  traveled  the  same  a» 
at  first,  A  would  have  traveled  twice 
as  far  as  B.  How  far  did  each  trav- 
el?        Ans.,  A  10  mi.;   B  14  mi. 

15.  A  starts  to  walk  from  P  to 
Q  at  the  rate  of  4  miles  an  hour, 
and  1  hour  later  B  starts  from  P  and 
overtakes  A  in  4  hours.  Walking 
on,  B  arrives  at  Q  2  hours  before  A. 
Find  the  distance  from   P  to  Q. 

Ans.,  60  mi. 

16.  It  is  11  miles  from  San  Jose 
to  Los  Gatos  A  carry-all  leaves  San 
Jose  at"  8  o'clock  to  go  to  Los  Gatos 
and  return  and  travels  5  miles  an 
hour.  When  must  a  carriage  which 
travels  8  miles  an  hour  leave  San 
Jose  that  it  may  meet  the  carry-all 
and  return  to  San  Jose  by  12 
o'clock?  Ans.,  10:40. 

17.  It  is  24  miles  from  San  Jose 
to  the  Lick  Observatory.  A  car- 
riage leaves  San  Jose  at  9  a.  m.  and 
travels  up  hill  at  the  rate  of  4  miles 
an  hour,  stops  an  hour  at  the  ob- 
servatory, and  returns  at  the  rate  of 
10  miles  an  hour.  A  second  car- 
riage leaves  San  Jose  at  3  p.  m.  and 
travels  at  the  rate  of  5  miles  an 
hour.  How  far  from  San  Jose  will 
the  carriages  pass  each  other? 

Ans.,  11^  mi. 

18.  A  man  can  row  up  stream  5 
miles  an  hour  and  down  the  stream 
7  miles  an  hour.  How  far  can  he 
row  up  stream  and  back  in  5  hours" 

Ans.,  103^  mi. 

19.  A  started  at  8  a.  m.  around 
a  mile  track,  walking  at  the  rate  of 
5  miles  an  hour  B  started  at  8:05 
and  walked  in  the  same  direction  at 
the  rate  of  4  miles  per  hour.  If  A 
rests  one  minute  at  the  end  of  each 
mile,  when  will  they  first  be  to- 
gether  again?  Ans.,  9  o'clock. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 

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